New chat PYQ Library Doubts Rank report

Sign in to see Recents

Sign in to save progress →

Differential Equations question

2023 · Shift 2 · Q18

Guest · try one question or challengeGuest · try one question or challenge

PYQ Library
JEE Advanced
Mathematics
Differential Equations
2023 · Shift 2 · Q18

Differential Equations question

2023 · Shift 2 · Q18

JEE AdvancedMathematicsDifferential EquationsMCQ+3 / −1

Let

f:[1, \infty) \rightarrow \mathbb{R}

be a differentiable function such that

f(1)=\frac{1}{3}

and

3 \int\limits_1^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 of

f(e)

is :

A
$\frac{e^2+4}{3}$
B
$\frac{\log _e 4+e}{3}$
C
$\frac{4 e^2}{3}$
D
$\frac{e^2-4}{3}$

More from Differential Equations

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 · 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 · 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 · 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 · 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 · 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 · MCQ
If $g (x) = \int_{s i n x}^{s i n (2 x)} sin^{- 1} (t) d t$ , then2017 · 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 · Multiple correct