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Differential Equations

23 questions · Mathematics · JEE Advanced

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PYQ Library
JEE Advanced
Mathematics
Differential Equations

Differential Equations

23 questions · Mathematics · JEE Advanced

For all x > 0, let y₁(x), y₂(x), and y₃(x) be the functions satisfying $\frac{d y _{1}}{d x} - (sin x)^{2} y_{1} = 0, y_{1} (1) = 5, \frac{d y _{2}}{d x} - (cos x)^{2} y_{2} = 0, y_{2} (1) = \frac{1}{3}, \frac{d y _{3}}{d x} - \frac{( 2 - x ^{3} )}{x ^{3}} y_{3} = 0, y_{3} (1) = \frac{3}{5 e},$ …2025 · Shift 1 · Q29 · Numerical
Let $y (x)$ be the solution of the differential equation $x^{2} \frac{d y}{d x} + x y = x^{2} + y^{2}, x > \frac{1}{e}$ satisfying $y (1) = 0$ . Then the value of $2 \frac{( y ( e ) ) ^{2}}{y ( e ^{2} )}$ is .2025 · Shift 2 · Q25 · Numerical
Let $f (x)$ be a continuously differentiable function on the interval $(0, \infty)$ such that $f (1) = 2$ and $t \to x lim \frac{t ^{10} f ( x ) - x ^{10} f ( t )}{t ^{9} - x ^{9}} = 1$ for each $x > 0$ . Then, for all $x > 0, f (x)$ is equal…2024 · Shift 1 · Q18 · MCQ
Let $f : [1, \infty) \to R$ be a differentiable function such that $f (1) = \frac{1}{3}$ and $3 1 \int x f (t) d t = x f (x) - \frac{x ^{3}}{3}, x \in [1, \infty)$ . Let $e$ denote the base of the natural logarithm. Then the value…2023 · Shift 2 · Q18 · MCQ
For $x \in R$ , let $y (x)$ be a solution of the differential equation $(x^{2} - 5) \frac{d y}{d x} - 2 x y = - 2 x (x^{2} - 5)^{2}$ such that $y (2) = 7$ . Then the maximum value of the function $y (x)$ is :2023 · Shift 2 · Q26 · Numerical
If $y (x)$ is the solution of the differential equation $x d y - (y^{2} - 4 y) d x = 0 for x > 0, y (1) = 2,$ and the slope of the curve $y = y (x)$ is never zero, then the value of $10 y (2)$ is2022 · Shift 2 · Q20 · Numerical
For $x \in R$ , let the function $y (x)$ be the solution of the differential equation $\frac{d y}{d x} + 12 y = cos (\frac{π}{12} x), y (0) = 0$ Then, which of the following statements is/are TRUE ?2022 · Shift 2 · Q32 · Multiple correct
Let $Γ$ denote a curve y = y(x) which is in the first quadrant and let the point (1, 0) lie on it. Let the tangent to I` at a point P intersect the y-axis at YP. If PYP has length 1 for each point P on I`, then which of the following…2019 · Shift 1 · Q23 · Multiple correct
Let f : R $\to$ R be a differentiable function with f(0) = 0. If y = f(x) satisfies the differential equation $\frac{d y}{d x} = (2 + 5 y) (5 y - 2)$ , then the value of $n \to - \infty lim f (x)$ is ...........2018 · Shift 2 · Q28 · Numerical
If y = y(x) satisfies the differential equation $8 x (9 + x) d y = (4 + 9 + x)^{- 1}$ dx, x > 0 and y(0) = $7$ , then y(256) =2017 · Shift 2 · Q20 · MCQ
If $g (x) = \int_{s i n x}^{s i n (2 x)} sin^{- 1} (t) d t$ , then2017 · Shift 2 · Q31 · Multiple correct
A solution curve of the differential equation $(x^{2} + x y + 4 x + 2 y + 4) \frac{d y}{d x} - y^{2} = 0,$ $x > 0,$ passes through the point $(1, 3)$ . Then the solution curve2016 · Shift 1 · Q23 · Multiple correct
Let $f : (0, \infty) \to R$ be a differentiable function such that $f^{'} (x) = 2 - \frac{f ( x )}{x}$ for all $x \in (0, \infty)$ and $f (1) e 1$ . Then2016 · Shift 1 · Q32 · Multiple correct
Let $y (x)$ be a solution of the differential equation $(1 + e^{x}) y^{'} + y e^{x} = 1.$ If $y (0) = 2$ , then which of the following statement is (are) true?2015 · Shift 1 · Q28 · Multiple correct
Consider the family of all circles whose centres lie on the straight line $y = x,$ If this family of circle is represented by the differential equation $P y^{''} + Q y^{'} + 1 = 0,$ where $x, y$ are functions of $y^{'}$ and $y^{'}$ $(h er e y^{'} = \frac{d y}{d x}, y^{''} = \frac{d ^{2} y}{d x ^{2}})$ …2015 · Shift 1 · Q29 · Multiple correct
The function $y = f (x)$ is the solution of the differential equation $\frac{d y}{d x} + \frac{x y}{x ^{2} - 1} = \frac{x ^{4} + 2 x}{1 - x ^{2}}$ in $(- 1, 1)$ satisfying $f (0) = 0$ . Then $- \frac{3}{2} \int \frac{3}{2} f (x) d (x)$ …2014 · Shift 2 · Q33 · MCQ
A curve passes through the point $(1, \frac{π}{6})$ . Let the slope of the curve at each point $(x, y)$ be $\frac{y}{x} + sec (\frac{y}{x}), x > 0.$ Then the equation of the curve is2013 · Shift 1 · Q36 · MCQ
If $y (x)$ satisfies the differential equation $y^{'} - y t an x = 2 x sec x$ and $y (0) = 0,$ then2012 · Shift 1 · Q34 · Multiple correct
Let $f : [1, \infty) \to [2, \infty)$ be a differentiable function such that $f (1) = 2$ . If $6 1 \int x f (t) d t = 3 x f (x) - x^{3} - 5$ for all $x \geq 1$ , then the value of f(2) is .2011 · Shift 1 · Q46 · Numerical
Let $y^{'} (x) + y (x) g^{'} (x) = g (x), g^{'} (x), y (0) = 0, x \in R,$ where $f^{'} (x)$ denotes $\frac{df ( x )}{d x}$ and $g (x)$ is a given non-constant…2011 · Shift 2 · Q32 · Numerical
Match the statements/expressions in Column I with the open intervals in Column II : Includes table2009 · Shift 1 · Q36 · MCQ
Match the statements/expressions in Column I with the values given in Column II: Includes table2009 · Shift 2 · Q22 · MCQ
Let a solution $y = y (x)$ of the differential equation, $x x^{2} - 1 d y - y y^{2} - 1 d x = 0$ satify $y (2) = \frac{2}{3} .$ STATEMENT-1 : $y (x) = sec (sec^{- 1} x - \frac{π}{6})$ …2008 · Shift 2 · Q29 · MCQ