List of top Mathematics Questions on Differential equations asked in JEE Main

If x=f(y) x = f(y) is the solution of the differential equation (1+y2)+(x−2etan⁡−1y)dydx=0,y∈(−π2,π2), (1 + y^2) + (x - 2e^{\tan^{-1}y}) \frac{dy}{dx} = 0, \quad y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right), with f(0)=1 f(0) = 1 , then f(13) f\left( \frac{1}{\sqrt{3}} \right) is equal to:
Let y=f(x) y = f(x) be the solution of the differential equationdydx+xyx2−1=x6+4x1−x2,−1<x<1\frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^6 + 4x}{\sqrt{1 - x^2}}, \quad -1 < x < 1 such that f(0)=0 f(0) = 0 . If 6∫−1/21/2f(x)dx=2π−α6 \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha then α2 \alpha^2 is equal to __________.

Let f:R→R f : \mathbb{R} \to \mathbb{R} be a twice differentiable function such that f(x+y)=f(x)f(y) f(x + y) = f(x) f(y) for all x,y∈R x, y \in \mathbb{R} . If f′(0)=4a f'(0) = 4a and f f satisfies f′′(x)−3af′(x)−f(x)=0 f''(x) - 3a f'(x) - f(x) = 0 , where a>0 a > 0 , then the area of the region R = {(x, y) | 0 ≤\leq y ≤\leq f(ax), 0 ≤\leq x ≤\leq 2\  is :

Let f(x) f(x) be a real differentiable function such that f(0)=1 f(0) = 1 and f(x+y)=f(x)f′(y)+f′(x)f(y) f(x + y) = f(x)f'(y) + f'(x)f(y) for all x,y∈R x, y \in \mathbb{R} . Then ∑n=1100log⁡ef(n) \sum_{n=1}^{100} \log_e f(n) is equal to :
Let x=x(y) x = x(y) be the solution of the differential equation y2dx+(x−1y)dy=0 y^2 dx + \left( x - \frac{1}{y} \right) dy = 0 . If x(1)=1 x(1) = 1 , then x(12) x\left( \frac{1}{2} \right) is :
Let y=y(x) y = y(x) be the solution of the differential equation cos⁡(xlog⁡(cos⁡x))2 dy+(sin⁡x−3sin⁡xlog⁡(cos⁡x)) dx=0,x∈(0,π2) \cos(x \log(\cos x))^2 \, dy + (\sin x - 3 \sin x \log(\cos x)) \, dx = 0, \quad x \in \left( 0, \frac{\pi}{2} \right) If y(π4)=−1 y\left( \frac{\pi}{4} \right) = -1 , then y(π6) y\left( \frac{\pi}{6} \right) is equal to:
If y=y(x) y = y(x) is the solution curve of the differential equation (x2−4) dy−(y2−3y) dx=0, (x^2 - 4) \, dy - (y^2 - 3y) \, dx = 0, with x>2, y(4)=32 x > 2, \, y(4) = \frac{3}{2} and the slope of the curve is never zero, then the value of y(10) y(10) equals: 
Let the solution y=y(x) y = y(x) of the differential equation dydx−y=1+4sin⁡x\frac{dy}{dx} - y = 1 + 4 \sin x satisfy y(π)=1 y(\pi) = 1 . Then y(π2)+10 y\left( \frac{\pi}{2} \right) + 10 is equal to _____
If the solution y=y(x) y = y(x) of the differential equation (x4+2x3+3x2+2x+2)dy−(2x2+2x+3)dx=0 \left( x^4 + 2x^3 + 3x^2 + 2x + 2 \right) dy - \left( 2x^2 + 2x + 3 \right) dx = 0
satisfies y(−1)=−π4 y(-1) = -\frac{\pi}{4} , then y(0) y(0) is equal to:
Let ∫0x1−(y′(t))2 dt=∫0xy(t) dt,0≤x≤3, y≥0, y(0)=0. \int_{0}^{x} \sqrt{1 - (y'(t))^2} \, dt = \int_{0}^{x} y(t) \, dt, \quad 0 \leq x \leq 3, \, y \geq 0, \, y(0) = 0. Then, at x=2 x = 2 , y′′+y+1 y'' + y + 1 is equal to:
For a differentiable function f:R→R f : \mathbb{R} \to \mathbb{R} , suppose f′(x)=3f(x)+α, f'(x) = 3f(x) + \alpha, where α∈R \alpha \in \mathbb{R} , f(0)=1 f(0) = 1 , and lim⁡x→−∞f(x)=7. \lim_{x \to -\infty} f(x) = 7. Then 9f(−log⁡23) 9f(-\log_2 3) is equal to __________ .
If log⁡ey=3sin⁡−1x \log_e y = 3 \sin^{-1}x , then (1−x)2y′′−xy′ (1 - x)^2 y'' - xy' at x=12 x = \frac{1}{2} is equal to:
If x=x(t) x = x(t) is the solution of the differential equation (t+1)dx=(2x+(t+1)4)dt,x(0)=2,(t + 1) dx = \left(2x + (t + 1)^4\right) dt, \quad x(0) = 2, then x(1) x(1) equals ____.  
Let f:R−{0}→R f : \mathbb{R} - \{0\} \rightarrow \mathbb{R} be a function satisfyingf(xy)=f(x)f(y)f\left( \frac{x}{y} \right) = \frac{f(x)}{f(y)}for all x,y x, y , f(y)≠0 f(y) \neq 0 . If f′(1)=2024 f'(1) = 2024 ,then
Let y=y(x) y = y(x) be the solution of the differential equation (1−x2) dy=[xy+(x3+2)3(1−x2)]dx (1 - x^2) \, dy = \left[ xy + \left( x^3 + 2 \right) \sqrt{3 \left( 1 - x^2 \right)} \right] dx , −1<x<1,y(0)=0 -1 < x < 1, y(0) = 0 . If y(12)=mn y\left( \frac{1}{2} \right) = \frac{m}{n} , m m and n n are coprime numbers, then m+n m + n is equal to _________\_\_\_\_\_\_\_\_\_.
Let y=y(x)y = y(x) be the solution of the differential equation sec⁡(x)dydx+[2(1−x)tan⁡(x)+x(2−x)]=0\sec(x) \frac{dy}{dx} + [2(1 - x) \tan(x) + x(2 - x)] = 0 such that y(0)=2y(0) = 2. Then y(2)y(2) is equal to:
If (t+1)dx=(2x+(t+1)3)dt(t+1)dx=(2x+(t+1)^3)dt and x(0)=2x(0)=2, then x(1)x(1) is equal to:
Let vertex A(2,3,1),B(3,2,−1),C(−2,1,3)A(2, 3, 1), B(3, 2, -1), C(-2, 1, 3). If ADAD is angle bisector of angle AA, then projection of AD→\overrightarrow{AD} on AC→\overrightarrow{AC} is equal to:
If dydx−(sin2x)(1+cos2x)y=sinx1+cos2x\frac {dy}{dx} - \frac {(sin2x)}{(1+cos^2x)}y = \frac {sinx}{1+cos^2x }and y(0)=0y(0) =0 then y(π2)y(\frac \pi2) is ………..
The solution of differential equation ydxdy=x(logex−logey+1), x>0,y>0y\frac {dx}{dy}=x(log_e x-log_e y+1),\ x>0, y>0 and passing through (e,1)(e,1) is
A function y=f(x)y=f(x) satisfiesf(x)sin2x+sinx−(1+cos2x)f′(x)=0f (x)sin2x+sinx-(1+cos^2x) f'(x)=0 with conditionf(0)=0f(0)=0.Thenf(π2)f(\frac{\pi}{2}) equals to
If the solution curve y=y x y = y \, x of the differential equation (1+y2)(1+log⁡ex)dx+x dy=0,x>0(1 + y^2) \left(1 + \log_e x\right) dx + x \, dy = 0, \quad x > 0 passes through the point (1,1) (1, 1) andy(e)=α−tan⁡(32)β+tan⁡(32),y(e) = \frac{\alpha - \tan\left(\frac{3}{2}\right)}{\beta + \tan\left(\frac{3}{2}\right)},then α+2β \alpha + 2\beta is
Let y=y(x)y = y(x) be the solution of the differential equation (1+y2)etan⁡xdx+cos⁡2x(1+e2tan⁡x)dy=0(1 + y^2)e^{\tan x} dx + \cos^2 x (1 + e^{2\tan x}) dy = 0, y(0)=1y(0) = 1. Then y(π4)y\left(\frac{\pi}{4}\right) is equal to:
Let y=y(x)y = y(x) be the solution curve of the differential equation sec⁡ydydx+2xsin⁡y=x3cos⁡y, \sec y \frac{dy}{dx} + 2x \sin y = x^3 \cos y, y(1)=0y(1) = 0. Then y(3)y\left(\sqrt{3}\right) is equal to:
The solution of the differential equation (x2+y2)dx−5xy dy=0, y(1)=0 (x^2 + y^2) dx - 5xy \, dy = 0, \, y(1) = 0 , is: