Differential Equations — JEE Main PYQs

243 previous year questions on Differential Equations (Mathematics), newest first. Open any question for its complete step-by-step solution.

Let $y = y (x)$ be the solution curve of the differential equation $(1 + x^{2}) d y + (y - tan^{- 1} x) d x = 0$ , $y (0) = 1$ . Then the value of $y (1)$ is :
JEE Main 2026View solution →
Let $y = y (x)$ be the solution of the differential equation $sec x \frac{dy}{dx} - 2 y = 2 + 3 sin x, x \in (- \frac{π}{2}, \frac{π}{2})$ , $y (0) = - \frac{7}{4}$ . Then $y (\frac{π}{6})$ is equal to:
JEE Main 2026View solution →
Let the solution curve of the differential equation $x d y - y d x = x^{2} + y^{2} d x, x > 0, y (1) = 0$ , be $y = y (x)$ . Then $y (3)$ is equal to
JEE Main 2026View solution →
If $y = y (x)$ satisfies the differential equation
$16 (x + 9 x) (4 + 9 + x) cos y d y = (1 + 2 sin y) d x$ , $x > 0$ and $y (256) = \frac{π}{2}$ , $y (49) = α$ , then $2 sin α$ is equal to :
JEE Main 2026View solution →
Let $y = y (x)$ be the solution of the differential equation
$x^{4} d y + (4 x^{3} y + 2 sin x) d x = 0$ , $x > 0$ , $y (\frac{π}{2}) = 0$ .

Then $π^{4} y (\frac{π}{3})$ is equal to :
JEE Main 2026View solution →
If the solution curve $y = f (x)$ of the differential equation $(x^{2} - 4) y^{'} - 2 x y + 2 x (4 - x^{2})^{2} = 0, x > 2$ , passes through the point (3,15), then the local maximum value of f is
JEE Main 2026View solution →
Let $y = y (x)$ be the solution of the differential equation
$x \frac{d y}{d x} - sin 2 y = x^{3} (2 - x^{3}) cos^{2} y, x \neq = 0$ .
If $y (2) = 0$ , then $tan (y (1))$ is equal to
JEE Main 2026View solution →
Let $y = y (x)$ be the solution of the differential equation $x \frac{d y}{d x} - y = x^{2} cot x, x \in (0, π)$ .
If $y (\frac{π}{2}) = \frac{π}{2}$ , then $6 y (\frac{π}{6}) - 8 y (\frac{π}{4})$ is equal to :
JEE Main 2026View solution →
If the curve $y = f (x)$ passes through the point $(1, e)$ and satisfies the differential equation $d y = y (2 + lo g_{e} x) d x$ , $x > 0$ , then $f (e)$ is equal to :
JEE Main 2026View solution →
Let $y = y (x)$ be the solution curve of the differential equation $(1 + sin x) \frac{d y}{d x} + (y + 1) cos x = 0$ , $y (0) = 0$ . If the curve $y = y (x)$ passes through the point $(α, \frac{- 1}{2})$ , then a value of $α$ is :
JEE Main 2026View solution →
Let $y = y (x)$ be the solution of the differential equation :
$\frac{d y}{d x} + (\frac{6 x ^{2} + ( 3 x ^{2} + 2 x ^{3} + 4 ) e ^{- 2 x}}{( x ^{3} + 2 ) ( 2 + e ^{- 2 x} )}) y = 2 + e^{- 2 x}$

$x \in (- 1, 2)$ , satisfying $y (0) = \frac{3}{2}$ .
If …
JEE Main 2026View solution →
Let $y = y (x)$ be the solution of the differential equation $\frac{d y}{d x} = (1 + x + x^{2}) (1 - y + y^{2}), y (0) = \frac{1}{2}$ . Then $(2 y (1) - 1)$ is equal to
JEE Main 2026View solution →
Let $y = y (x)$ be the solution of the differential equation
$x sin (\frac{y}{x}) d y = (y sin (\frac{y}{x}) - x) d x, y (1) = \frac{π}{2}$ and let $α = cos (\frac{y ( e ^{12} )}{e ^{12}})$ . Then the number of integral value of $p$ , f …
JEE Main 2026View solution →
Let $y = y (x)$ be the solution of the differential equation $(tan x)^{1/2} d y = (sec^{3} x - (tan x)^{3/2} y) d x$ , $0 < x < \frac{π}{2}$ , $y (\frac{π}{4}) = \frac{6 2}{5}$ . If $y (\frac{π}{3}) = \frac{4}{5} α$ , then $α^{4}$ equals ______
JEE Main 2026View solution →
Let $f : R \to R$ be such that $f (x y) = f (x) f (y)$ , for all $x, y \in R$ and $f (0) \neq = 0$ . Let $g : [1, \infty) \to R$ be a differentiable function such that
$x^{2} g (x) = \int_{1}^{x} (t^{2} f (t) - tg (t)) d t$
Then $g (2)$ is equal to :
JEE Main 2026View solution →
Let $y = y (x)$ be the solution of the differential equation
$(x^{2} - x x^{2} - 1) d y + (y (x - x^{2} - 1) - x) d x = 0$ , $x \geq 1$ . If $y (1) = 1$ , then the greatest integer less than $y (5)$ is ______.
JEE Main 2026View solution →
Let $y = y (x)$ be the solution of the differential equation
$x 1 - x^{2} d y + (y 1 - x^{2} - x cos^{- 1} x) d x = 0$ ,
$x \in (0, 1), lim_{x \to 1^{-}} y (x) = 1$ . Then $y (\frac{1}{2})$ equals :
JEE Main 2026View solution →
Let $x = x (y)$ be the solution of the differential equation $2 y^{2} \frac{d x}{d y} - 2 x y + x^{2} = 0, y > 1, x (e) = e$ . Then $x (e^{2})$ is equal to:
JEE Main 2026View solution →
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 $f^{''} (x) - 3 a f^{'} (x) - f (x) = 0$ , $a > 0$ , then the area of the region …
JEE Main 2025View solution →
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 :
JEE Main 2025View solution →
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 …
JEE Main 2025View solution →
If $x = f (y)$ is the solution of the differential equation
$(1 + y^{2}) + (x - 2 e^{t a n^{- 1} y}) \frac{dy}{dx} = 0, y \in (- \frac{π}{2}, \frac{π}{2})$
with …
JEE Main 2025View solution →
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 $α^{2}$ is equal to ____.
JEE Main 2025View solution →
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 …
JEE Main 2025View solution →
Let $x = x (y)$ be the solution of the differential equation $y = (x - y \frac{dx}{dy}) sin (\frac{x}{y}), y > 0$ and $x (1) = \frac{π}{2}$ .

Then $cos (x (2))$ is equal to :
JEE Main 2025View solution →
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 $f (\frac{π}{4})$ is equal to:
JEE Main 2025View solution →
Let $y = y (x)$ be the solution of the differential equation $cos x (lo g_{e} (cos x))^{2} d y + (sin x - 3 y sin x lo g_{e} (cos x)) d x = 0$ , $x \in (0, \frac{π}{2})$ . If $y (\frac{π}{4}) = \frac{- 1}{l o g _{e} 2}$ , then …
JEE Main 2025View solution →
Let $f : (0, \infty) \to R$ be a twice differentiable function. If for some $a \neq = 0, \int_{0}^{1} f (λ x) d λ = af (x)$ , $f (1) = 1$ and $f (16) = \frac{1}{8}$ , then …
JEE Main 2025View solution →
If $y = y (x)$ is the solution of the differential equation,
$4 - x^{2} \frac{d y}{d x} = ((sin^{- 1} (\frac{x}{2}))^{2} - y) sin^{- 1} (\frac{x}{2})$ ,
$- 2 \leq x \leq 2, y (2) = (\frac{π ^{2} - 8}{4})$ , then $y^{2} (0)$ is equal to …
JEE Main 2025View solution →
Let $f : (0, \infty) \to R$ be a function which is differentiable at all points of its domain and satisfies the condition $x^{2} f^{'} (x) = 2 xf (x) + 3$ , with $f (1) = 4$ . Then $2 f (2)$ is equal to:
JEE Main 2025View solution →
Let $y = y (x)$ be the solution of the differential equation $2 cos x \frac{dy}{dx} = sin 2 x - 4 y sin x, x \in (0, \frac{π}{2})$ . If $y (\frac{π}{3}) = 0$ , then …
JEE Main 2025View solution →
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 to
JEE Main 2025View solution →
Let f be a differentiable function such that
$2 (x + 2)^{2} f (x) - 3 (x + 2)^{2} = 10 \int_{0}^{x} (t + 2) f (t) dt$ , $x \geq 0$ . Then $f (2)$ is equal to ____.
JEE Main 2025View solution →
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 $12 (y (\frac{π}{4}) - e^{- 2})$ is equal to ______.
JEE Main 2025View solution →
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
JEE Main 2025View solution →
Let $f : R \to R$ be a thrice differentiable odd function satisfying $f^{'} (x) \geq 0, f^{''} (x) = f (x), f (0) = 0, f^{'} (0) = 3$ . Then $9 f (lo g_{e} 3)$ is equal to ______.
JEE Main 2025View solution →
Let g be a differentiable function such that $\int_{0}^{x} g (t) dt = x - \int_{0}^{x} tg (t) dt$ , $x \geq 0$ and let $y = y (x)$ satisfy the differential equation …
JEE Main 2025View solution →
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 $x = 2$ , the value of cosy is:
JEE Main 2025View solution →
Let $f : [0, \infty) \to R$ be differentiable function such that $f (x) = 1 - 2 x + \int_{0}^{x} e^{x - t} f (t) dt$ for all $x \in [0, \infty)$ . Then the area of the region bounded by $y = f (x)$ and the coordinate a …
JEE Main 2025View solution →
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 $\int_{- 3}^{3} y (x) dx$ is :
JEE Main 2025View solution →
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 :
JEE Main 2025View solution →
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 $y (1)$ is :-
JEE Main 2025View solution →
Let for some function y = f(x), ∫₀ˣ t·f(t) dt = x²·f(x), x > 0 and f(2) = 3. Then f(6) is equal to:
JEE Main 2025View solution →
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
JEE Main 2024View solution →
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 ____.
JEE Main 2024View solution →
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 to
JEE Main 2024View solution →
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 …
JEE Main 2024View solution →
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:
JEE Main 2024View solution →
The differential equation of the family of circles passing through the origin and having centre at the line $y = x$ is
JEE Main 2024View solution →
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 to
JEE Main 2024View solution →