Differential Equations

208 questions · Mathematics · JEE Main

Differential Equations

208 questions · Mathematics · JEE Main

Let $y = y (x)$ be the solution of the differential equation $\frac{d y}{d x} + 2 y sec^{2} x = 2 sec^{2} x + 3 tan x \cdot sec^{2} x$ such that $y (0) = \frac{5}{4}$ . Then $12 (y (\frac{π}{4}) - e^{- 2})$ …2025 · 2 Apr · Shift 2 · Q49 · Numerical
Let $g$ be a differentiable function such that $\int_{0}^{x} g (t) d t = x - \int_{0}^{x} tg (t) d t, x \geq 0$ and let $y = y (x)$ satisfy the differential equation $\frac{d y}{d x} - y tan x = 2 (x + 1) sec xg (x), x \in [0, \frac{π}{2})$ …2025 · 3 Apr · Shift 1 · Q39 · MCQ
Let $y = y (x)$ be the solution of the differential equation $\frac{d y}{d x} + 3 (tan^{2} x) y + 3 y = sec^{2} x, y (0) = \frac{1}{3} + e^{3}$ . Then $y (\frac{π}{4})$ is equal to :2025 · 3 Apr · Shift 2 · Q33 · MCQ
If a curve $y = y (x)$ passes through the point $(1, \frac{π}{2})$ and satisfies the differential equation $(7 x^{4} cot y - e^{x} cosec y) \frac{d x}{d y} = x^{5}, x \geq 1$ , then at $x = 2$ …2025 · 4 Apr · Shift 2 · Q32 · MCQ
Let $y = y (x)$ be the solution curve of the differential equation $x (x^{2} + e^{x}) d y + (e^{x} (x - 2) y - x^{3}) d x = 0, x > 0$ , passing through the point $(1, 0)$ . Then $y (2)$ is equal to :2025 · 7 Apr · Shift 1 · Q32 · MCQ
Let y = y(x) be the solution of the differential equation $(x^{2} + 1) y^{'} - 2 x y = (x^{4} + 2 x^{2} + 1) cos x$ , $y (0) = 1$ . Then $- 3 \int 3 y (x) d x$ is :2025 · 7 Apr · Shift 2 · Q27 · MCQ
Let $f (x) = x - 1$ and $g (x) = e^{x}$ for $x \in R$ . If $\frac{d y}{d x} = (e^{- 2 x} g (f (f (x))) - \frac{y}{x})$ , $y (0) = 0$ , then $y (1)$ is2025 · 8 Apr · Shift 2 · Q27 · MCQ
Let $x = x (y)$ be the solution of the differential equation $y^{2} d x + (x - \frac{1}{y}) d y = 0$ . If $x (1) = 1$ , then $x (\frac{1}{2})$ is :2025 · 22 Jan · Shift 1 · Q34 · MCQ
Let $f (x)$ be a real differentiable function such that $f (0) = 1$ and $f (x + y) = f (x) f^{'} (y) + f^{'} (x) f (y)$ for all $x, y \in R$ . Then $\sum_{n = 1}^{100} lo g_{e} f (n)$ is equal to :2025 · 22 Jan · Shift 1 · Q36 · MCQ
Let $f : R \to R$ be a twice differentiable function such that $f (x + y) = f (x) f (y)$ for all $x, y \in R$ . If $f^{'} (0) = 4 a$ and $f$ satisfies $f^{''} (x) - 3 a f^{'} (x) - f (x) = 0, a > 0$ …2025 · 22 Jan · Shift 1 · Q43 · MCQ
If $x = f (y)$ is the solution of the differential equation $(1 + y^{2}) + (x - 2 e^{t a n^{- 1} y}) \frac{d y}{d x} = 0, y \in (- \frac{π}{2}, \frac{π}{2})$ with $f (0) = 1$ , then $f (\frac{1}{3})$ …2025 · 22 Jan · Shift 2 · Q32 · MCQ
Let $y = f (x)$ be the solution of the differential equation $\frac{d y}{d x} + \frac{x y}{x ^{2} - 1} = \frac{x ^{6} + 4 x}{1 - x ^{2}}, - 1 < x < 1$ such that $f (0) = 0$ . If $6 \int_{- 1/2}^{1/2} f (x) d x = 2 π - α$ then…2025 · 22 Jan · Shift 2 · Q47 · Numerical
Let a curve $y = f (x)$ pass through the points $(0, 5)$ and $(lo g_{e} 2, k)$ . If the curve satisfies the differential equation $2 (3 + y) e^{2 x} d x - (7 + e^{2 x}) d y = 0$ , then $k$ is equal to2025 · 23 Jan · Shift 1 · Q26 · MCQ
Let $x = x (y)$ be the solution of the differential equation $y = (x - y \frac{d x}{d y}) sin (\frac{x}{y}), y > 0$ and $x (1) = \frac{π}{2}$ . Then $cos (x (2))$ is equal to :2025 · 23 Jan · Shift 2 · Q45 · MCQ
Let $y = y (x)$ be the solution of the differential equation $(x y - 5 x^{2} 1 + x^{2}) d x + (1 + x^{2}) d y = 0, y (0) = 0$ . Then $y (3)$ is equal to2025 · 24 Jan · Shift 1 · Q34 · MCQ
Let $f$ be a differentiable function such that $2 (x + 2)^{2} f (x) - 3 (x + 2)^{2} = 10 \int_{0}^{x} (t + 2) f (t) d t, x \geq 0$ . Then $f (2)$ is equal to .2025 · 24 Jan · Shift 1 · Q47 · Numerical
Let $y = y (x)$ be the solution of the differential equation $2 cos x \frac{d y}{d x} = sin 2 x - 4 y sin x, x \in (0, \frac{π}{2})$ . If $y (\frac{π}{3}) = 0$ , then $y^{'} (\frac{π}{4}) + y (\frac{π}{4})$ …2025 · 24 Jan · Shift 2 · Q48 · Numerical
Let for some function $y = f (x), \int_{0}^{x} t f (t) d t = x^{2} f (x), x > 0$ and $f (2) = 3$ . Then $f (6)$ is equal to2025 · 28 Jan · Shift 1 · Q39 · MCQ
If $y = y (x)$ is the solution of the differential equation, $4 - x^{2} \frac{d y}{d x} = ((sin^{- 1} (\frac{x}{2}))^{2} - y) sin^{- 1} (\frac{x}{2}), - 2 \leq x \leq 2, y (2) = \frac{π ^{2} - 8}{4}$ …2025 · 28 Jan · Shift 2 · Q48 · Numerical
Let y = y(x) be the solution of the differential equation : $cos x (lo g_{e} (cos x))^{2} d y + (sin x - 3 y sin x lo g_{e} (cos x)) d x = 0$ , x ∈ (0, $\frac{π}{2}$ ). If $y (\frac{π}{4})$ = $- \frac{1}{l o g _{e} 2}$ , then $y (\frac{π}{6})$ …2025 · 29 Jan · Shift 1 · Q36 · MCQ
If for the solution curve $y = f (x)$ of the differential equation $\frac{d y}{d x} + (tan x) y = \frac{2 + s e c x}{( 1 + 2 s e c x ) ^{2}}$ , $x \in (\frac{- π}{2}, \frac{π}{2}), f (\frac{π}{3}) = \frac{3}{10}$ , then $f (\frac{π}{4})$ …2025 · 29 Jan · Shift 2 · Q38 · MCQ
Let $y = y (x)$ be the solution of the differential equation $\frac{d y}{d x} = 2 x (x + y)^{3} - x (x + y) - 1, y (0) = 1$ . Then, $(\frac{1}{2} + y (\frac{1}{2}))^{2}$ equals :2024 · 1 Feb · Shift 1 · Q44 · MCQ
If $x = x (t)$ is the solution of the differential equation $(t + 1) d x = (2 x + (t + 1)^{4}) dt, x (0) = 2$ , then, $x (1)$ equals .2024 · 1 Feb · Shift 1 · Q51 · Numerical
Let $α$ be a non-zero real number. Suppose $f : R \to R$ is a differentiable function such that $f (0) = 2$ and $x \to - \infty lim f (x) = 1$ . If $f^{'} (x) = α f (x) + 3$ , for all $x \in R$ …2024 · 1 Feb · Shift 2 · Q44 · MCQ
If $\frac{d x}{d y} = \frac{1 + x - y ^{2}}{y}, x (1) = 1$ , then $5 x (2)$ is equal to .2024 · 1 Feb · Shift 2 · Q54 · Numerical
If the solution $y = y (x)$ of the differential equation $(x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 2) d y - (2 x^{2} + 2 x + 3) d x = 0$ satisfies $y (- 1) = - \frac{π}{4}$ , then $y (0)$ is equal to :2024 · 4 Apr · Shift 1 · Q43 · MCQ
Let the solution $y = y (x)$ of the differential equation $\frac{d y}{d x} - y = 1 + 4 sin x$ satisfy $y (π) = 1$ . Then $y (\frac{π}{2}) + 10$ is equal to .2024 · 4 Apr · Shift 1 · Q51 · Numerical
Let $y = y (x)$ be the solution of the differential equation $(x^{2} + 4)^{2} d y + (2 x^{3} y + 8 x y - 2) d x = 0$ . If $y (0) = 0$ , then $y (2)$ is equal to2024 · 4 Apr · Shift 2 · Q41 · MCQ
Let $y = y (x)$ be the solution of the differential equation $(x + y + 2)^{2} d x = d y, y (0) = - 2$ . Let the maximum and minimum values of the function $y = y (x)$ in $[0, \frac{π}{3}]$ be $α$ and $β$ , respectively. If $(3 α + π)^{2} + β^{2} = γ + δ 3, γ, δ \in Z$ …2024 · 4 Apr · Shift 2 · Q57 · Numerical
If $y = y (x)$ is the solution of the differential equation $\frac{d y}{d x} + 2 y = sin (2 x), y (0) = \frac{3}{4}$ , then $y (\frac{π}{8})$ is equal to :2024 · 5 Apr · Shift 1 · Q50 · MCQ
The differential equation of the family of circles passing through the origin and having centre at the line $y = x$ is :2024 · 5 Apr · Shift 2 · Q50 · MCQ
Let $y = y (x)$ be the solution of the differential equation $\frac{d y}{d x} + \frac{2 x}{( 1 + x ^{2} ) ^{2}} y = x e^{\frac{1}{( 1 + x ^{2} )}}; y (0) = 0.$ Then the area enclosed by the curve $f (x) = y (x) e^{- \frac{1}{( 1 + x ^{2} )}}$ …2024 · 5 Apr · Shift 2 · Q51 · Numerical
Let $y = y (x)$ be the solution of the differential equation $(2 x lo g_{e} x) \frac{d y}{d x} + 2 y = \frac{3}{x} lo g_{e} x, x > 0$ and $y (e^{- 1}) = 0$ . Then, $y (e)$ is equal to2024 · 6 Apr · Shift 1 · Q40 · MCQ
Let $y = y (x)$ be the solution of the differential equation $(1 + x^{2}) \frac{d y}{d x} + y = e^{t a n^{- 1} x}$ , $y (1) = 0$ . Then $y (0)$ is2024 · 6 Apr · Shift 1 · Q46 · MCQ
Suppose the solution of the differential equation $\frac{d y}{d x} = \frac{( 2 + α ) x - β y + 2}{β x - 2 α y - ( β γ - 4 α )}$ represents a circle passing through origin. Then the radius of this circle is :2024 · 6 Apr · Shift 2 · Q42 · MCQ
If the solution $y (x)$ of the given differential equation $(e^{y} + 1) cos x d x + e^{y} sin x d y = 0$ passes through the point $(\frac{π}{2}, 0)$ , then the value of $e^{y (\frac{π}{6})}$ …2024 · 6 Apr · Shift 2 · Q53 · Numerical
Let $f (x)$ be a positive function such that the area bounded by $y = f (x), y = 0$ from $x = 0$ to $x = a > 0$ is $e^{- a} + 4 a^{2} + a - 1$ . Then the differential equation, whose general solution is $y = c_{1} f (x) + c_{2}$ , where $c_{1}$ and $c_{2}$ are arbitrary…2024 · 8 Apr · Shift 1 · Q36 · MCQ
Let $y = y (x)$ be the solution of the differential equation $(1 + y^{2}) e^{t a n x} d x + cos^{2} x (1 + e^{2 t a n x}) d y = 0, y (0) = 1$ . Then $y (\frac{π}{4})$ is equal to2024 · 8 Apr · Shift 1 · Q39 · MCQ
Let $y = y (x)$ be the solution curve of the differential equation $sec y \frac{d y}{d x} + 2 x sin y = x^{3} cos y, y (1) = 0$ . Then $y (3)$ is equal to:2024 · 8 Apr · Shift 2 · Q31 · MCQ
Let $α ∣ x ∣ = ∣ y ∣ e^{x y - β}, α, β \in N$ be the solution of the differential equation $x d y - y d x + x y (x d y + y d x) = 0, y (1) = 2$ . Then $α + β$ is equal to …2024 · 8 Apr · Shift 2 · Q57 · Numerical
The solution of the differential equation $(x^{2} + y^{2}) d x - 5 x y d y = 0, y (1) = 0$ , is :2024 · 9 Apr · Shift 1 · Q35 · MCQ
The solution curve, of the differential equation $2 y \frac{d y}{d x} + 3 = 5 \frac{d y}{d x}$ , passing through the point $(0, 1)$ is a conic, whose vertex lies on the line :2024 · 9 Apr · Shift 1 · Q46 · MCQ
Let $0 \int x 1 - (y^{'} (t))^{2} d t = \int_{0}^{x} y (t) d t, 0 \leq x \leq 3, y \geq 0, y (0) = 0$ . Then at $x = 2, y^{''} + y + 1$ is equal to2024 · 9 Apr · Shift 2 · Q34 · MCQ
For a differentiable function $f : R \to R$ , suppose $f^{'} (x) = 3 f (x) + α$ , where $α \in R, f (0) = 1$ and $lim_{x \to - \infty} f (x) = 7$ . Then $9 f (- lo g_{e} 3)$ is equal…2024 · 9 Apr · Shift 2 · Q54 · Numerical
Let $x = x (t)$ and $y = y (t)$ be solutions of the differential equations $\frac{d x}{dt} + a x = 0$ and $\frac{d y}{dt} + by = 0$ respectively, $a, b \in R$ …2024 · 27 Jan · Shift 1 · Q36 · MCQ
If the solution of the differential equation $(2 x + 3 y - 2) d x + (4 x + 6 y - 7) d y = 0, y (0) = 3$ , is $α x + β y + 3 lo g_{e} ∣2 x + 3 y - γ ∣ = 6$ , then $α + 2 β + 3 γ$ is equal to .2024 · 27 Jan · Shift 1 · Q52 · Numerical
If $y = y (x)$ is the solution curve of the differential equation $(x^{2} - 4) d y - (y^{2} - 3 y) d x = 0, x > 2, y (4) = \frac{3}{2}$ and the slope of the curve is never zero, then the value of $y (10)$ equals :2024 · 27 Jan · Shift 2 · Q42 · MCQ
If the solution curve, of the differential equation $\frac{d y}{d x} = \frac{x + y - 2}{x - y}$ passing through the point $(2, 1)$ is $tan^{- 1} (\frac{y - 1}{x - 1}) - \frac{1}{β} lo g_{e} (α + (\frac{y - 1}{x - 1})^{2}) = lo g_{e} ∣ x - 1∣$ …2024 · 27 Jan · Shift 2 · Q60 · Numerical
A function $y = f (x)$ satisfies $f (x) sin 2 x + sin x - (1 + cos^{2} x) f^{'} (x) = 0$ with condition $f (0) = 0$ . Then, $f (\frac{π}{2})$ is equal to2024 · 29 Jan · Shift 1 · Q49 · MCQ
If the solution curve $y = y (x)$ of the differential equation $(1 + y^{2}) (1 + lo g_{e} x) d x + x d y = 0, x > 0$ passes through the point $(1, 1)$ and $y (e) = \frac{α - t a n ( \frac{3}{2} )}{β + t a n ( \frac{3}{2} )}$ …2024 · 29 Jan · Shift 1 · Q60 · Numerical
If $sin (\frac{y}{x}) = lo g_{e} ∣ x ∣ + \frac{α}{2}$ is the solution of the differential equation $x cos (\frac{y}{x}) \frac{d y}{d x} = y cos (\frac{y}{x}) + x$ and $y (1) = \frac{π}{3}$ , then $α^{2}$ …2024 · 29 Jan · Shift 2 · Q42 · MCQ
Let $y = y (x)$ be the solution of the differential equation $sec x d y + {2 (1 - x) tan x + x (2 - x)} d x = 0$ such that $y (0) = 2$ . Then $y (2)$ is equal to:2024 · 30 Jan · Shift 1 · Q45 · MCQ
Let $y = y (x)$ be the solution of the differential equation $(1 - x^{2}) d y = [x y + (x^{3} + 2) 3 (1 - x^{2})] d x, - 1 < x < 1, y (0) = 0$ . If $y (\frac{1}{2}) = \frac{m}{n}, m$ …2024 · 30 Jan · Shift 1 · Q52 · Numerical
Let $Y = Y (X)$ be a curve lying in the first quadrant such that the area enclosed by the line $Y - y = Y^{'} (x) (X - x)$ and the co-ordinate axes, where $(x, y)$ is any point on the curve, is always $\frac{- y ^{2}}{2 Y ^{'} ( x )} + 1, Y^{'} (x) e q 0$ …2024 · 30 Jan · Shift 2 · Q52 · Numerical
Let $y = y (x)$ be the solution of the differential equation $\frac{d y}{d x} = \frac{( t a n x ) + y}{s i n x ( s e c x - s i n x t a n x )}, x \in (0, \frac{π}{2})$ satisfying the condition $y (\frac{π}{4}) = 2$ . Then, $y (\frac{π}{3})$ …2024 · 31 Jan · Shift 1 · Q37 · MCQ
The solution curve of the differential equation $y \frac{d x}{d y} = x (lo g_{e} x - lo g_{e} y + 1), x > 0, y > 0$ passing through the point $(e, 1)$ is2024 · 31 Jan · Shift 1 · Q46 · MCQ
The temperature $T (t)$ of a body at time $t = 0$ is $16 0^{\circ} F$ and it decreases continuously as per the differential equation $\frac{d T}{d t} = - K (T - 80)$ , where $K$ is a positive constant. If $T (15) = 12 0^{\circ} F$ , then…2024 · 31 Jan · Shift 2 · Q42 · MCQ
Let $y = y (x)$ be the solution of the differential equation $sec^{2} x d x + (e^{2 y} tan^{2} x + tan x) d y = 0, 0 < x < \frac{π}{2}, y (π /4) = 0$ . If $y (π /6) = α$ , then $e^{8 α}$ is equal to …2024 · 31 Jan · Shift 2 · Q52 · Numerical
The area enclosed by the closed curve $C$ given by the differential equation $\frac{d y}{d x} + \frac{x + a}{y - 2} = 0, y (1) = 0$ is $4 π$ . Let $P$ and $Q$ be the points of intersection of the curve $C$ and the $y$ -axis. If…2023 · 1 Feb · Shift 1 · Q32 · MCQ
If $y = y (x)$ is the solution curve of the differential equation $\frac{d y}{d x} + y tan x = x sec x, 0 \leq x \leq \frac{π}{3}, y (0) = 1$ , then $y (\frac{π}{6})$ is equal to2023 · 1 Feb · Shift 1 · Q34 · MCQ
Let $α x = exp (x^{β} y^{γ})$ be the solution of the differential equation $2 x^{2} y d y - (1 - x y^{2}) d x = 0, x > 0, y (2) = lo g_{e} 2$ . Then $α + β - γ$ equals :2023 · 1 Feb · Shift 2 · Q39 · MCQ
Let $y = y (x)$ be a solution of the differential equation $(x cos x) d y + (x y sin x + y cos x - 1) d x = 0, 0 < x < \frac{π}{2}$ . If $\frac{π}{3} y (\frac{π}{3}) = 3$ , then $\frac{π}{6} y^{''} (\frac{π}{6}) + 2 y^{'} (\frac{π}{6})$ …2023 · 6 Apr · Shift 1 · Q36 · Numerical
If the solution curve $f (x, y) = 0$ of the differential equation $(1 + lo g_{e} x) \frac{d x}{d y} - x lo g_{e} x = e^{y}, x > 0$ , passes through the points $(1, 0)$ and $(α, 2)$ , then $α^{α}$ is equal to :2023 · 6 Apr · Shift 2 · Q30 · MCQ
If the solution curve of the differential equation $(y - 2 lo g_{e} x) d x + (x lo g_{e} x^{2}) d y = 0, x > 1$ passes through the points $(e, \frac{4}{3})$ and $(e^{4}, α)$ , then $α$ …2023 · 8 Apr · Shift 1 · Q41 · Numerical
Let the solution curve $x = x (y), 0 < y < \frac{π}{2}$ , of the differential equation $(lo g_{e} (cos y))^{2} cos y d x - (1 + 3 x lo g_{e} (cos y)) sin y d y = 0$ satisfy $x (\frac{π}{3}) = \frac{1}{2 l o g _{e} 2}$ …2023 · 8 Apr · Shift 2 · Q40 · Numerical
Let $f$ be a differentiable function such that $x^{2} f (x) - x = 4 0 \int x t f (t) d t$ , $f (1) = \frac{2}{3}$ . Then $18 f (3)$ is equal to :2023 · 10 Apr · Shift 1 · Q36 · MCQ
Let the tangent at any point P on a curve passing through the points (1, 1) and $(\frac{1}{10}, 100)$ , intersect positive $x$ -axis and $y$ -axis at the points A and B respectively. If $PA : PB = 1 : k$ and…2023 · 10 Apr · Shift 2 · Q33 · Numerical
Let $y = y (x)$ be a solution curve of the differential equation. $(1 - x^{2} y^{2}) d x = y d x + x d y$ . If the line $x = 1$ intersects the curve $y = y (x)$ at $y = 2$ and the line $x = 2$ intersects the curve $y = y (x)$ at $y = α$ , then a…2023 · 11 Apr · Shift 1 · Q29 · MCQ
Let $y = y (x)$ be the solution of the differential equation $\frac{d y}{d x} + \frac{5}{x ( x ^{5} + 1 )} y = \frac{( x ^{5} + 1 ) ^{2}}{x ^{7}}, x > 0$ . If $y (1) = 2$ , then $y (2)$ is equal to :2023 · 11 Apr · Shift 2 · Q34 · MCQ
Let $y = y (x), y > 0$ , be a solution curve of the differential equation $(1 + x^{2}) d y = y (x - y) d x$ . If $y (0) = 1$ and $y (22) = β$ , then2023 · 12 Apr · Shift 1 · Q25 · MCQ
Let $y = y_{1} (x)$ and $y = y_{2} (x)$ be the solution curves of the differential equation $\frac{d y}{d x} = y + 7$ with initial conditions $y_{1} (0) = 0$ and $y_{2} (0) = 1$ respectively. Then the curves $y = y_{1} (x)$ and $y = y_{2} (x)$ intersect at2023 · 13 Apr · Shift 1 · Q27 · MCQ
If $y = y (x)$ is the solution of the differential equation $\frac{d y}{d x} + \frac{4 x}{( x ^{2} - 1 )} y = \frac{x + 2}{( x ^{2} - 1 ) ^{\frac{5}{2}}}, x > 1$ such that $y (2) = \frac{2}{9} lo g_{e} (2 + 3) and y (2) = α lo g_{e} (α + β) + β - γ, α, β, γ \in N, then α β γ is equal to$ …2023 · 13 Apr · Shift 2 · Q41 · Numerical
Let $x = x (y)$ be the solution of the differential equation $2 (y + 2) lo g_{e} (y + 2) d x + (x + 4 - 2 lo g_{e} (y + 2)) d y = 0, y > - 1$ with $x (e^{4} - 2) = 1$ . Then $x (e^{9} - 2)$ is equal to :2023 · 15 Apr · Shift 1 · Q24 · MCQ
Let $y = y (x)$ be the solution of the differential equation $x^{3} d y + (x y - 1) d x = 0, x > 0, y (\frac{1}{2}) = 3 - e$ . Then y (1) is equal to2023 · 24 Jan · Shift 1 · Q29 · MCQ
Let $y = y (x)$ be the solution of the differential equation $(x^{2} - 3 y^{2}) d x + 3 x y d y = 0, y (1) = 1$ . Then $6 y^{2} (e)$ is equal to2023 · 24 Jan · Shift 2 · Q30 · MCQ
Let $y = y (x)$ be the solution curve of the differential equation $\frac{d y}{d x} = \frac{y}{x} (1 + x y^{2} (1 + lo g_{e} x)), x > 0, y (1) = 3$ . Then $\frac{y ^{2} ( x )}{9}$ is equal to :2023 · 25 Jan · Shift 1 · Q37 · MCQ
Let $y = y (t)$ be a solution of the differential equation $\frac{d y}{d t} + α y = γ e^{- β t}$ where, $α > 0, β > 0$ and $γ > 0$ . Then $t \to \infty lim y (t)$ 2023 · 25 Jan · Shift 2 · Q35 · MCQ
Let $y = f (x)$ be the solution of the differential equation $y (x + 1) d x - x^{2} d y = 0, y (1) = e$ . Then $x \to 0^{+} lim f (x)$ is equal to2023 · 29 Jan · Shift 1 · Q32 · MCQ
Let $y = y (x)$ be the solution of the differential equation $x lo g_{e} x \frac{d y}{d x} + y = x^{2} lo g_{e} x, (x > 1)$ . If $y (2) = 2$ , then $y (e)$ is equal to2023 · 29 Jan · Shift 2 · Q24 · MCQ
Let the solution curve $y = y (x)$ of the differential equation $\frac{d y}{d x} - \frac{3 x ^{5} t a n ^{- 1} ( x ^{3} )}{( 1 + x ^{6} ) ^{3/2}} y = 2 x exp {\frac{x ^{3} - t a n ^{- 1} x ^{3}}{( 1 + x ^{6} )}} pass through the origin. Then y (1) is equal to :$ …2023 · 30 Jan · Shift 1 · Q23 · MCQ
The solution of the differential equation $\frac{d y}{d x} = - (\frac{x ^{2} + 3 y ^{2}}{3 x ^{2} + y ^{2}}), y (1) = 0$ is :2023 · 30 Jan · Shift 2 · Q34 · MCQ
Let $y = y (x)$ be the solution of the differential equation $(3 y^{2} - 5 x^{2}) y d x + 2 x (x^{2} - y^{2}) d y = 0$ such that $y (1) = 1$ . Then $(y (2))^{3} - 12 y (2)$ is equal to :2023 · 31 Jan · Shift 2 · Q36 · MCQ
If x = x(y) is the solution of the differential equation $y \frac{d x}{d y} = 2 x + y^{3} (y + 1) e^{y}, x (1) = 0$ ; then x(e) is equal to :2022 · 24 Jun · Shift 1 · Q32 · MCQ
The slope of the tangent to a curve $C : y = y (x)$ at any point $(x, y)$ on it is $\frac{2 e ^{2 x} - 6 e ^{- x} + 9}{2 + 9 e ^{- 2 x}}$ . If $C$ passes through the points $$\left(0, \frac{1}{2}+\frac{\pi}{2…2022 · 25 Jul · Shift 1 · Q30 · MCQ
The general solution of the differential equation $(x - y^{2}) d x + y (5 x + y^{2}) d y = 0$ is :2022 · 25 Jul · Shift 1 · Q31 · MCQ
Let $y = y (x)$ be the solution of the differential equation $\frac{d y}{d x} = \frac{4 y ^{3} + 2 y x ^{2}}{3 x y ^{2} + x ^{3}}, y (1) = 1$ . If for some $n \in N, y (2) \in [n - 1, n)$ , then $n$ is equal to .2022 · 25 Jul · Shift 2 · Q41 · Numerical
Let $g : (0, \infty) \to R$ be a differentiable function such that $\int (\frac{x ( c o s x - s i n x )}{e ^{x} + 1} + \frac{g ( x ) ( e ^{x} + 1 - x e ^{x} )}{( e ^{x} + 1 ) ^{2}}) d x = \frac{x g ( x )}{e ^{x} + 1} + c$ …2022 · 25 Jun · Shift 1 · Q32 · MCQ
Let $y = y (x)$ be the solution of the differential equation $(x + 1) y^{'} - y = e^{3 x} (x + 1)^{2}$ , with $y (0) = \frac{1}{3}$ . Then, the point $x = - \frac{4}{3}$ for the curve $y = y (x)$ is :2022 · 25 Jun · Shift 1 · Q35 · MCQ
If the solution curve $y = y (x)$ of the differential equation $y^{2} d x + (x^{2} - x y + y^{2}) d y = 0$ , which passes through the point (1, 1) and intersects the line $y = 3 x$ at the point $(α, 3 α)$ , then value of $lo g_{e} (3 α)$ …2022 · 25 Jun · Shift 1 · Q36 · MCQ
If $y = y (x)$ is the solution of the differential equation $2 x^{2} \frac{d y}{d x} - 2 x y + 3 y^{2} = 0$ such that $y (e) = \frac{e}{3}$ , then y(1) is equal to :2022 · 25 Jun · Shift 2 · Q32 · MCQ
If $\frac{d y}{d x} + 2 y tan x = sin x, 0 < x < \frac{π}{2}$ and $y (\frac{π}{3}) = 0$ , then the maximum value of $y (x)$ is :2022 · 26 Jul · Shift 1 · Q33 · MCQ
Let the solution curve $y = f (x)$ of the differential equation $\frac{d y}{d x} + \frac{x y}{x ^{2} - 1} = \frac{x ^{4} + 2 x}{1 - x ^{2}}$ , $x \in (- 1, 1)$ pass through the origin. Then $- \frac{3}{2} \int \frac{3}{2} f (x) d x$ …2022 · 26 Jul · Shift 2 · Q28 · MCQ
Suppose $y = y (x)$ be the solution curve to the differential equation $\frac{d y}{d x} - y = 2 - e^{- x}$ such that $x \to \infty lim y (x)$ is finite. If $a$ and $b$ are respectively the $x$ - and $y$ -intercepts of the tangent to…2022 · 26 Jul · Shift 2 · Q37 · Numerical
Let the solution curve y = y(x) of the differential equation $(4 + x^{2}) d y - 2 x (x^{2} + 3 y + 4) d x = 0$ pass through the origin. Then y(2) is equal to .2022 · 26 Jun · Shift 1 · Q36 · Numerical
Let $S = (0, 2 π) - {\frac{π}{2}, \frac{3 π}{4}, \frac{3 π}{2}, \frac{7 π}{4}}$ . Let $y = y (x)$ , x $\in$ S, be the solution curve of the differential equation $\frac{d y}{d x} = \frac{1}{1 + s i n 2 x}, y (\frac{π}{4}) = \frac{1}{2}$ …2022 · 26 Jun · Shift 1 · Q41 · Numerical
If $y = y (x)$ is the solution of the differential equation $x \frac{d y}{d x} + 2 y = x e^{x}$ , $y (1) = 0$ then the local maximum value of the function $z (x) = x^{2} y (x) - e^{x}, x \in R$ is :2022 · 26 Jun · Shift 2 · Q32 · MCQ
If the solution of the differential equation $\frac{d y}{d x} + e^{x} (x^{2} - 2) y = (x^{2} - 2 x) (x^{2} - 2) e^{2 x}$ satisfies $y (0) = 0$ , then the value of y(2) is…2022 · 26 Jun · Shift 2 · Q33 · MCQ
Let $y = y_{1} (x)$ and $y = y_{2} (x)$ be two distinct solutions of the differential equation $\frac{d y}{d x} = x + y$ , with $y_{1} (0) = 0$ and $y_{2} (0) = 1$ respectively. Then, the number of points of intersection of $y = y_{1} (x)$ and $y = y_{2} (x)$ is2022 · 27 Jul · Shift 1 · Q33 · MCQ
Let $y = y (x)$ be the solution curve of the differential equation $sin (2 x^{2}) lo g_{e} (tan x^{2}) d y + (4 x y - 42 x sin (x^{2} - \frac{π}{4})) d x = 0$ , $0 < x < \frac{π}{2}$ …2022 · 27 Jul · Shift 1 · Q42 · Numerical
If $\frac{d y}{d x} + \frac{2 ^{x - y} ( 2 ^{y} - 1 )}{2 ^{x} - 1} = 0$ , x, y > 0, y(1) = 1, then y(2) is equal to :2022 · 27 Jun · Shift 1 · Q28 · MCQ
If the solution curve of the differential equation $((tan^{- 1} y) - x) d y = (1 + y^{2}) d x$ passes through the point (1, 0), then the abscissa of the point on the curve whose ordinate is tan(1), is2022 · 27 Jun · Shift 2 · Q28 · MCQ
Let $y = y (x)$ be the solution of the differential equation $(1 - x^{2}) d y = (x y + (x^{3} + 2) 1 - x^{2}) d x, - 1$ -$$1 is equal to $\underline{\hspace{2cm}}$.2022 · 27 Jun · Shift 2 · Q43 · Numerical
Let the solution curve of the differential equation $x d y = (x^{2} + y^{2} + y) d x, x > 0$ , intersect the line $x = 1$ at $y = 0$ and the line $x = 2$ at $y = α$ . Then the value of $α$ is :2022 · 28 Jul · Shift 1 · Q23 · MCQ
If $y = y (x), x \in (0, π /2)$ be the solution curve of the differential equation $(sin^{2} 2 x) \frac{d y}{d x} + (8 sin^{2} 2 x + 2 sin 4 x) y = 2 e^{- 4 x} (2 sin 2 x + cos 2 x)$ , with $y (π /4) = e^{- π}$ …2022 · 28 Jul · Shift 1 · Q31 · MCQ
Let $y = y (x)$ be the solution curve of the differential equation $\frac{d y}{d x} + \frac{1}{x ^{2} - 1} y = (\frac{x - 1}{x + 1})^{1/2}$ , $x > 1$ passing through the point $(2, \frac{1}{3})$ . Then $\sqrt{7}\,…2022 · 28 Jul · Shift 2 · Q31 · MCQ
Let the solution curve $y = y (x)$ of the differential equation $[\frac{x}{x ^{2} - y ^{2}} + e^{\frac{y}{x}}] x \frac{d y}{d x} = x + [\frac{x}{x ^{2} - y ^{2}} + e^{\frac{y}{x}}] y$ …2022 · 28 Jun · Shift 1 · Q31 · MCQ
Let y = y(x) be the solution of the differential equation $x (1 - x^{2}) \frac{d y}{d x} + (3 x^{2} y - y - 4 x^{3}) = 0$ , $x > 1$ , with $y (2) = - 2$ . Then y(3) is equal to :2022 · 28 Jun · Shift 1 · Q32 · MCQ
Let x = x(y) be the solution of the differential equation $2 y e^{x / y^{2}} d x + (y^{2} - 4 x e^{x / y^{2}}) d y = 0$ such that x(1) = 0. Then, x(e) is equal to :2022 · 28 Jun · Shift 2 · Q31 · MCQ
Let the slope of the tangent to a curve y = f(x) at (x, y) be given by 2 $tan x (cos x - y)$ . If the curve passes through the point $(\frac{π}{4}, 0)$ , then the value of $0 \int π /2 y d x$ is equal to :2022 · 28 Jun · Shift 2 · Q32 · MCQ
Let the solution curve $y = y (x)$ of the differential equation $(1 + e^{2 x}) (\frac{d y}{d x} + y) = 1$ pass through the point $(0, \frac{π}{2})$ . Then, $x \to \infty lim e^{x} y (x)$ …2022 · 29 Jul · Shift 1 · Q31 · MCQ
If the solution curve of the differential equation $\frac{d y}{d x} = \frac{x + y - 2}{x - y}$ passes through the points $(2, 1)$ and $(k + 1, 2), k > 0$ , then2022 · 29 Jul · Shift 2 · Q29 · MCQ
Let $y = y (x)$ be the solution curve of the differential equation $\frac{d y}{d x} + (\frac{2 x ^{2} + 11 x + 13}{x ^{3} + 6 x ^{2} + 11 x + 6}) y = \frac{( x + 3 )}{x + 1}, x > - 1$ , which passes through the point $(0, 1)$ . Then $y (1)$ is equal to :2022 · 29 Jul · Shift 2 · Q30 · MCQ
Let the solution curve of the differential equation $x \frac{d y}{d x} - y = y^{2} + 16 x^{2}$ , $y (1) = 3$ be $y = y (x)$ . Then y(2) is equal to:2022 · 29 Jun · Shift 1 · Q21 · MCQ
Let y = y(x) be the solution of the differential equation $${{dy} \over {dx}} + {{\sqrt 2 y} \over {2{{\cos }^4}x - {{\cos }^2}x}} = x{e^{{{\tan }^{ - 1}}(\sqrt 2 \cot 2x)}},\,0 2 is equal to $\underline{\hspace{2cm}}$.2022 · 29 Jun · Shift 1 · Q37 · Numerical
If y = y(x) is the solution of the differential equation $(1 + e^{2 x}) \frac{d y}{d x} + 2 (1 + y^{2}) e^{x} = 0$ and y (0) = 0, then $6 (y^{'} (0) + (y (lo g_{e} 3))^{2})$ …2022 · 29 Jun · Shift 2 · Q29 · MCQ
Let y = y(x), x > 1, be the solution of the differential equation $(x - 1) \frac{d y}{d x} + 2 x y = \frac{1}{x - 1}$ , with $y (2) = \frac{1 + e ^{4}}{2 e ^{4}}$ . If $y (3) = \frac{e ^{α} + 1}{β e ^{α}}$ , then the…2022 · 29 Jun · Shift 2 · Q36 · Numerical
If y = y(x) is the solution curve of the differential equation $x^{2} d y + (y - \frac{1}{x}) d x = 0$ ; x > 0 and y(1) = 1, then $y (\frac{1}{2})$ is equal to :2021 · 1 Sep · Shift 2 · Q28 · MCQ
If y = y(x) is the solution of the differential equation, $\frac{d y}{d x} + 2 y tan x = sin x, y (\frac{π}{3}) = 0$ , then the maximum value of the function y(x) over R is equal to:2021 · 16 Mar · Shift 1 · Q36 · MCQ
Let the curve y = y(x) be the solution of the differential equation, $\frac{d y}{d x}$ = 2(x + 1). If the numerical value of area bounded by the curve y = y(x) and x-axis is $\frac{4 8}{3}$ , then the value of y(1) is equal to …2021 · 16 Mar · Shift 1 · Q37 · Numerical
If y = y(x) is the solution of the differential equation $\frac{d y}{d x}$ + (tan x) y = sin x, $0 \leq x \leq \frac{π}{3}$ , with y(0) = 0, then $y (\frac{π}{4})$ equal to :2021 · 16 Mar · Shift 2 · Q24 · MCQ
Let C1 be the curve obtained by the solution of differential equation $2 x y \frac{d y}{d x} = y^{2} - x^{2}, x > 0$ . Let the curve C2 be the solution of $\frac{2 x y}{x ^{2} - y ^{2}} = \frac{d y}{d x}$ . If both the curves pass through…2021 · 16 Mar · Shift 2 · Q31 · MCQ
Which of the following is true for y(x) that satisfies the differential equation $\frac{d y}{d x}$ = xy $-$ 1 + x $-$ y; y(0) = 0 :2021 · 17 Mar · Shift 1 · Q26 · MCQ
If the curve y = y(x) is the solution of the differential equation $2 (x^{2} + x^{5/4}) d y - y (x + x^{1/4}) d x = 2 x^{9/4} d x$ , x > 0 which passes through the point $(1, 1 - \frac{4}{3} lo g_{e} 2)$ , then the value…2021 · 17 Mar · Shift 2 · Q26 · MCQ
Let y = y(x) be the solution of the differential equation $cos x (3 sin x + cos x + 3) d y = (1 + y sin x (3 sin x + cos x + 3)) d x, 0 \leq x \leq \frac{π}{2}, y (0) = 0$ . Then, $y (\frac{π}{3})$ is equal to :2021 · 17 Mar · Shift 2 · Q33 · MCQ
Let y = y(x) be the solution of the differential equation $\frac{d y}{d x} = (y + 1) ((y + 1) e^{x^{2} /2} - x)$ , 0 < x < 2.1, with y(2) = 0. Then the value of $\frac{d y}{d x}$ at x = 1 is equal to :2021 · 18 Mar · Shift 2 · Q23 · MCQ
Let y = y(x) be the solution of the differential equation xdy $-$ ydx = $(x^{2} - y^{2}) d x$ , x $\geq$ 1, with y(1) = 0. If the area bounded by the line x = 1, x = e $π$ , y = 0 and y = y(x) is $α$ e2 $π$ + $β$ , then the…2021 · 18 Mar · Shift 2 · Q40 · Numerical
Let y = y(x) be the solution of the differential equation $x tan (\frac{y}{x}) d y = (y tan (\frac{y}{x}) - x) d x$ , $- 1 \leq x \leq 1$ , $y (\frac{1}{2}) = \frac{π}{6}$ . Then the…2021 · 20 Jul · Shift 1 · Q29 · MCQ
Let y = y(x) be the solution of the differential equation $e^{x} 1 - y^{2} d x + (\frac{y}{x}) d y = 0$ , y(1) = $-$ 1. Then the value of (y(3))2 is equal to :2021 · 20 Jul · Shift 1 · Q32 · MCQ
Let y = y(x) satisfies the equation $\frac{d y}{d x} - ∣ A ∣ = 0$ , for all x > 0, where $A = y 02 sin x - 1 0 11 \frac{1}{x}$ . If $y (π) = π + 2$ …2021 · 20 Jul · Shift 2 · Q33 · MCQ
Let a curve y = y(x) be given by the solution of the differential equation $cos (\frac{1}{2} cos^{- 1} (e^{- x})) d x = e^{2 x} - 1 d y$ . If it intersects y-axis at y = $-$ 1, and the intersection point of…2021 · 20 Jul · Shift 2 · Q40 · Numerical
Let y = y(x) be the solution of the differential equation $cos e c^{2} x d y + 2 d x = (1 + y cos 2 x) cos e c^{2} x d x$ , with $y (\frac{π}{4}) = 0$ . Then, the value of $(y (0) + 1)^{2}$ is equal to :2021 · 22 Jul · Shift 2 · Q26 · MCQ
Let y = y(x) be the solution of the differential equation $((x + 2) e^{(\frac{y + 1}{x + 2})} + (y + 1)) d x = (x + 2) d y$ , y(1) = 1. If the domain of y = y(x) is an open interval ( $α$ , $β$ ), then…2021 · 22 Jul · Shift 2 · Q46 · Numerical
The population P = P(t) at time 't' of a certain species follows the differential equation $\frac{d P}{d t}$ = 0.5P – 450. If P(0) = 850, then the time at which population becomes zero is :2021 · 24 Feb · Shift 1 · Q25 · MCQ
If a curve passes through the origin and the slope of the tangent to it at any point (x, y) is $\frac{x ^{2} - 4 x + y + 8}{x - 2}$ , then this curve also passes through the point :2021 · 25 Feb · Shift 1 · Q29 · MCQ
If the curve, y = y(x) represented by the solution of the differential equation (2xy2 $-$ y)dx + xdy = 0, passes through the intersection of the lines, 2x $-$ 3y = 1 and 3x + 2y = 8, then |y(1)| is equal to .2021 · 25 Feb · Shift 2 · Q48 · Numerical
Let y = y(x) be the solution of the differential equation $\frac{d y}{d x} = 1 + x e^{y - x}, - 2 < x < 2, y (0) = 0$ then, the minimum value of $y (x), x \in (- 2, 2)$ is equal to :2021 · 25 Jul · Shift 1 · Q30 · MCQ
Let y = y(x) be solution of the following differential equation $e^{y} \frac{d y}{d x} - 2 e^{y} sin x + sin x cos^{2} x = 0, y (\frac{π}{2}) = 0$ If $y (0) = lo g_{e} (α + β e^{- 2})$ , then $4(\alpha + \beta…2021 · 25 Jul · Shift 1 · Q38 · Numerical
Let y = y(x) be the solution of the differential equation xdy = (y + x3 cosx)dx with y( $π$ ) = 0, then $y (\frac{π}{2})$ is equal to :2021 · 25 Jul · Shift 2 · Q40 · MCQ
Let y = y(x) be a solution curve of the differential equation $(y + 1) tan^{2} x d x + tan x d y + y d x = 0$ , $x \in (0, \frac{π}{2})$ . If $x \to 0 + lim x y (x) = 1$ , then the value of $y (\frac{π}{4})$ …2021 · 26 Aug · Shift 1 · Q24 · MCQ
Let y(x) be the solution of the differential equation 2x2 dy + (ey $-$ 2x)dx = 0, x > 0. If y(e) = 1, then y(1) is equal to :2021 · 26 Aug · Shift 2 · Q27 · MCQ
The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at initial time t = 0. The number of bacteria is increased by 20% in 2 hours. If the population of bacteria is…2021 · 26 Feb · Shift 1 · Q35 · MCQ
The difference between degree and order of a differential equation that represents the family of curves given by $y^{2} = a (x + \frac{a}{2})$ , a > 0 is .2021 · 26 Feb · Shift 1 · Q41 · Numerical
If y = y(x) is the solution of the equation $e^{s i n y} cos y \frac{d y}{d x} + e^{s i n y} cos x = cos x$ , y(0) = 0; then $1 + y (\frac{π}{6}) + \frac{3}{2} y (\frac{π}{3}) + \frac{1}{2} y (\frac{π}{4})$ …2021 · 26 Feb · Shift 1 · Q42 · Numerical
Let y = y(x) be the solution of the differential equation $\frac{d y}{d x} = 2 (y + 2 sin x - 5) x - 2 cos x$ such that y(0) = 7. Then y( $π$ ) is equal to :2021 · 27 Aug · Shift 1 · Q27 · MCQ
Let us consider a curve, y = f(x) passing through the point ( $-$ 2, 2) and the slope of the tangent to the curve at any point (x, f(x)) is given by f(x) + xf'(x) = x2. Then :2021 · 27 Aug · Shift 1 · Q28 · MCQ
If the solution curve of the differential equation (2x $-$ 10y3)dy + ydx = 0, passes through the points (0, 1) and (2, $β$ ), then $β$ is a root of the equation :2021 · 27 Aug · Shift 2 · Q27 · MCQ
Let y = y(x) be solution of the differential equation $lo g (\frac{d y}{d x}) = 3 x + 4 y$ , with y(0) = 0. If $y (- \frac{2}{3} lo g_{e} 2) = α lo g_{e} 2$ , then the value of $α$ is equal…2021 · 27 Jul · Shift 1 · Q33 · MCQ
If $y = y (x), y \in [0, \frac{π}{2})$ is the solution of the differential equation $sec y \frac{d y}{d x} - sin (x + y) - sin (x - y) = 0$ , with y(0) = 0, then $5 y^{'} (\frac{π}{2})$ is equal to …2021 · 27 Jul · Shift 1 · Q46 · Numerical
Let y = y(x) be the solution of the differential equation (x $-$ x3)dy = (y + yx2 $-$ 3x4)dx, x > 2. If y(3) = 3, then y(4) is equal to :2021 · 27 Jul · Shift 2 · Q30 · MCQ
Let y = y(x) be the solution of the differential equation dy = e $α$ x + y dx; $α \in$ N. If y(loge2) = loge2 and y(0) = loge $(\frac{1}{2})$ , then the value of $α$ is equal to .2021 · 27 Jul · Shift 2 · Q41 · Numerical
If $\frac{d y}{d x} = \frac{2 ^{x + y} - 2 ^{x}}{2 ^{y}}$ , y(0) = 1, then y(1) is equal to :2021 · 31 Aug · Shift 1 · Q34 · MCQ
If $\frac{d y}{d x} = \frac{2 ^{x} y + 2 ^{y} .2 ^{x}}{2 ^{x} + 2 ^{x + y} l o g _{e} 2}$ , y(0) = 0, then for y = 1, the value of x lies in the interval :2021 · 31 Aug · Shift 2 · Q28 · MCQ
If $y \frac{d y}{d x} = x \frac{y ^{2}}{x ^{2}} + \frac{ϕ ( \frac{y ^{2}}{x ^{2}} )}{ϕ ^{'} ( \frac{y ^{2}}{x ^{2}} )}$ , x > 0, $ϕ$ > 0, and y(1) = $-$ 1, then $ϕ (\frac{y ^{2}}{4})$ …2021 · 31 Aug · Shift 2 · Q29 · MCQ
Let y = y(x) be the solution of the differential equation, $\frac{2 + s i n x}{y + 1} . \frac{d y}{d x} = - cos x$ , y > 0,y(0) = 1. If y( $π$ ) = a and $\frac{d y}{d x}$ at x = $π$ is b, then the ordered pair (a, b) is equal to…2020 · 2 Sep · Shift 1 · Q37 · MCQ
If a curve y = f(x), passing through the point (1, 2), is the solution of the differential equation, 2x2dy= (2xy + y2)dx, then $f (\frac{1}{2})$ is equal to :2020 · 2 Sep · Shift 2 · Q30 · MCQ
The solution curve of the differential equation, (1 + e-x)(1 + y2) $\frac{d y}{d x}$ = y2, which passes through the point (0, 1), is :2020 · 3 Sep · Shift 1 · Q22 · MCQ
If x3dy + xy dx = x2dy + 2y dx; y(2) = e and x > 1, then y(4) is equal to :2020 · 3 Sep · Shift 2 · Q32 · MCQ
Let y = y(x) be the solution of the differential equation, xy'- y = x2(xcosx + sinx), x > 0. if y ( $π$ ) = $π$ then $y^{''} (\frac{π}{2}) + y (\frac{π}{2})$ is equal to :2020 · 4 Sep · Shift 1 · Q38 · MCQ
The solution of the differential equation $\frac{d y}{d x} - \frac{y + 3 x}{l o g _{e} ( y + 3 x )} + 3 = 0$ is: (where c is a constant of integration)2020 · 4 Sep · Shift 2 · Q39 · MCQ
If y = y(x) is the solution of the differential equation $\frac{5 + e ^{x}}{2 + y} . \frac{d y}{d x} + e^{x} = 0$ satisfying y(0) = 1, then a value of y(loge13) is :2020 · 5 Sep · Shift 1 · Q39 · MCQ
Let y = y(x) be the solution of the differential equation cosx $\frac{d y}{d x}$ + 2ysinx = sin2x, x $\in$ $(0, \frac{π}{2})$ . If y $(\frac{π}{3})$ = 0, then y $(\frac{π}{4})$ is…2020 · 5 Sep · Shift 2 · Q20 · MCQ
The general solution of the differential equation $1 + x^{2} + y^{2} + x^{2} y^{2}$ + xy $\frac{d y}{d x}$ = 0 is : (where C is a constant of integration)2020 · 6 Sep · Shift 1 · Q21 · MCQ
If y = y(x) is the solution of the differential equation, $e^{y} (\frac{d y}{d x} - 1) = e^{x}$ such that y(0) = 0, then y(1) is equal to:2020 · 7 Jan · Shift 1 · Q35 · MCQ
Let y = y(x) be the solution curve of the differential equation, $(y^{2} - x) \frac{d y}{d x} = 1$ , satisfying y(0) = 1. This curve intersects the x-axis at a point whose abscissa is :2020 · 7 Jan · Shift 2 · Q37 · MCQ
Let y = y(x) be a solution of the differential equation, $1 - x^{2} \frac{d y}{d x} + 1 - y^{2} = 0$ , |x| < 1. If $y (\frac{1}{2}) = \frac{3}{2}$ , then $y (- \frac{1}{2})$ …2020 · 8 Jan · Shift 1 · Q31 · MCQ
If for x $\geq$ 0, y = y(x) is the solution of the differential equation (x + 1)dy = ((x + 1)2 + y – 3)dx, y(2) = 0, then y(3) is equal to .2020 · 9 Jan · Shift 1 · Q31 · Numerical
If $\frac{d y}{d x} = \frac{x y}{x ^{2} + y ^{2}}$ ; y(1) = 1; then a value of x satisfying y(x) = e is :2020 · 9 Jan · Shift 2 · Q27 · MCQ
Let y = y(x) be the solution of the differential equation, $(x^{2} + 1)^{2} \frac{d y}{d x} + 2 x (x^{2} + 1) y = 1$ such that y(0) = 0. If $a y (1)$ = $\frac{π}{32}$ , then the value of 'a' is :2019 · 8 Apr · Shift 1 · Q38 · MCQ
The solution of the differential equation $x \frac{d y}{d x} + 2 y$ = x2 (x $e$ 0) with y(1) = 1, is :2019 · 9 Apr · Shift 1 · Q41 · MCQ
If $cos x \frac{d y}{d x} - y sin x = 6 x$ , (0 < x < $\frac{π}{2}$ ) and $y (\frac{π}{3})$ = 0 then $y (\frac{π}{6})$ is equal to :-2019 · 9 Apr · Shift 2 · Q25 · MCQ
If y = y(x) is the solution of the differential equation, x $\frac{d y}{d x}$ + 2y = x2, satisfying y(1) = 1, then y( $\frac{1}{2}$ ) is equal to :2019 · 9 Jan · Shift 1 · Q30 · MCQ
Let f : [0,1] $\to$ R be such that f(xy) = f(x).f(y), for all x, y $\in$ [0, 1], and f(0) $e$ 0. If y = y(x) satiesfies the differential equation, $\frac{d y}{d x}$ = f(x) with y(0) = 1, then y $(\frac{1}{4})$ + y $(\frac{3}{4})$ …2019 · 9 Jan · Shift 2 · Q41 · MCQ
If y = y(x) is the solution of the differential equation $\frac{d y}{d x} = (tan x - y) sec^{2} x$ , $x \in (- \frac{π}{2}, \frac{π}{2})$ , such that y (0) = 0, then $y (- \frac{π}{4})$ …2019 · 10 Apr · Shift 1 · Q26 · MCQ
Let y = y(x) be the solution of the differential equation, $\frac{d y}{d x} + y tan x = 2 x + x^{2} tan x$ , $x \in (- \frac{π}{2}, \frac{π}{2})$ , such that y(0) = 1. Then :2019 · 10 Apr · Shift 2 · Q34 · MCQ
If $\frac{d y}{d x} + \frac{3}{c o s ^{2} x} y = \frac{1}{c o s ^{2} x}, x \in (\frac{- π}{3}, \frac{π}{3})$ and $y (\frac{π}{4}) = \frac{4}{3},$ then $y (- \frac{π}{4})$ …2019 · 10 Jan · Shift 1 · Q42 · MCQ
Let f be a differentiable function such that f '(x) = 7 - $\frac{3}{4} \frac{f ( x )}{x},$ (x > 0) and f(1) $e$ 4. Then $x \to 0^{'} lim$ xf $(\frac{1}{x})$ :2019 · 10 Jan · Shift 2 · Q24 · MCQ
The curve amongst the family of curves represented by the differential equation, (x2 – y2)dx + 2xy dy = 0 which passes through (1, 1) is :2019 · 10 Jan · Shift 2 · Q27 · MCQ
If y(x) is the solution of the differential equation $\frac{d y}{d x} + (\frac{2 x + 1}{x}) y = e^{- 2 x}, x > 0,$ where $y (1) = \frac{1}{2} e^{- 2},$ then2019 · 11 Jan · Shift 1 · Q24 · MCQ
The solution of the differential equation, $\frac{d y}{d x}$ = (x – y)2, when y(1) = 1, is :2019 · 11 Jan · Shift 2 · Q37 · MCQ
Consider the differential equation, $y^{2} d x + (x - \frac{1}{y}) d y = 0$ , If value of y is 1 when x = 1, then the value of x for which y = 2, is :2019 · 12 Apr · Shift 1 · Q41 · MCQ
The general solution of the differential equation (y2 – x3)dx – xydy = 0 (x $e$ 0) is : (where c is a constant of integration)2019 · 12 Apr · Shift 2 · Q25 · MCQ
Let y = y(x) be the solution of the differential equation, x $\frac{d y}{d x}$ + y = x loge x, (x > 1). If 2y(2) = loge 4 $-$ 1, then y(e) is equal to :2019 · 12 Jan · Shift 1 · Q45 · MCQ
If a curve passes through the point (1, –2) and has slope of the tangent at any point (x, y) on it as $\frac{x ^{2} - 2 y}{x}$ , then the curve also passes through the point :2019 · 12 Jan · Shift 2 · Q26 · MCQ
Let y = y(x) be the solution of the differential equation $\frac{d y}{d x} + 2 y = f (x),$ where $f (x) = {1, 0, x \in [0, 1] o t h er w i se$ …2018 · 15 Apr · Shift 1 · Q39 · MCQ
The curve satifying the differeial equation, (x2 $-$ y2) dx + 2xydy = 0 and passing through the point (1, 1) is :2018 · 15 Apr · Shift 2 · Q37 · MCQ
Let y = y(x) be the solution of the differential equation $sin x \frac{d y}{d x} + y cos x = 4 x$ , $x \in (0, π)$ . If $y (\frac{π}{2}) = 0$ , then $y (\frac{π}{6})$ is equal to :2018 · Shift 0 · Q38 · MCQ
The curve satisfying the differential equation, ydx $-$ (x + 3y2)dy = 0 and passing through the point (1, 1), also passes through the point :2017 · 8 Apr · Shift 1 · Q45 · MCQ
If 2x = y $^{\frac{1}{5}}$ + y $^{- \frac{1}{5}}$ and (x2 $-$ 1) $\frac{d ^{2} y}{d x ^{2}}$ + $λ$ x $\frac{d y}{d x}$ + ky = 0, then $λ$ + k is equal to :2017 · 9 Apr · Shift 1 · Q38 · MCQ
If $(2 + sin x) \frac{d y}{d x} + (y + 1) cos x = 0$ and y(0) = 1, then $y (\frac{π}{2})$ is equal to :2017 · Shift 0 · Q37 · MCQ
If f(x) is a differentiable function in the interval (0, $\infty$ ) such that f (1) = 1 and $t \to x lim$ $\frac{t ^{2} f ( x ) - x ^{2} f ( t )}{t - x} = 1,$ for each x > 0, then $f (\frac{3}{2})$ …2016 · 9 Apr · Shift 1 · Q41 · MCQ
The solution of the differential equation $\frac{d y}{d x} + \frac{y}{2} sec x = \frac{t a n x}{2 y},$ where 0 $\leq$ x < $\frac{π}{2}$ , and y (0) = 1, is given by :2016 · 10 Apr · Shift 1 · Q35 · MCQ
If a curve $y = f (x)$ passes through the point $(1, - 1)$ and satisfies the differential equation, $y (1 + x y) d x = x d y$ , then $f (- \frac{1}{2})$ is equal to :2016 · Shift 0 · Q25 · MCQ
Let $y (x)$ be the solution of the differential equation $(x lo g x) \frac{d y}{d x} + y = 2 x lo g x, (x \geq 1) .$ Then $y (e)$ is equal to :2015 · Shift 0 · Q28 · MCQ
Let the population of rabbits surviving at time $t$ be governed by the differential equation $\frac{d p ( t )}{d t} = \frac{1}{2} p (t) - 200.$ If $p (0) = 100,$ then $p (t)$ equals:2014 · Shift 0 · Q27 · MCQ
At present, a firm is manufacturing $2000$ items. It is estimated that the rate of change of production P w.r.t. additional number of workers $x$ is given by $\frac{d p}{d x} = 100 - 12 x .$ If the firm employs $25$ more workers,…2013 · Shift 0 · Q30 · MCQ
The population $p (t)$ at time $t$ of a certain mouse species satisfies the differential equation $\frac{d p ( t )}{d t} = 0.5 p (t) - 450.$ If $p (0) = 850,$ then the time at which the population becomes zero is…2012 · Shift 0 · Q30 · MCQ
If $\frac{d y}{d x} = y + 3 > 0$ and $y (0) = 2,$ then $y (ln 2)$ is equal to :2011 · Shift 0 · Q37 · MCQ
Let $I$ be the purchase value of an equipment and $V (t)$ be the value after it has been used for $t$ years. The value $V (t)$ depreciates at a rate given by differential equation $\frac{d v ( t )}{d t} = - k (T - t),$ …2011 · Shift 0 · Q38 · MCQ
Solution of the differential equation $cos x d y = y (sin x - y) d x, 0 < x < \frac{π}{2}$ is :2010 · Shift 0 · Q33 · MCQ
The solution of the differential equation $\frac{d y}{d x} = \frac{x + y}{x}$ satisfying the condition $y (1) = 1$ is :2008 · Shift 0 · Q35 · MCQ