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Differential Equations

JEE Mains Maths. Try each one first, then open its answer.

Top 10 JEE Mains PYQs to practise first

The questions and kinds of question asked most often come first. Each one says how often and in which papers.

  1. Question 1

    This type asked 41 times: 7 in 2021, 6 in 2022, 7 in 2023, 7 in 2024, 6 in 2025, 8 in 2026. This one: 8 Apr 2026 S2 See all 41 of this type

    67
    Let y=y(x)y=y(x) be the solution of the differential equation x1−x2dy+(y1−x2−xcos⁡−1x)dx=0x\sqrt{1 -x^{2}}dy+\left( y\sqrt{1 -x^{2}}-x\cos^{- 1}x \right)dx= 0, x∈(0,1),lim⁡x→1−y(x)=1x\in (0,1),\lim_{x\rightarrow1^{-}} y(x) = 1. Then y(12)y\left( \frac{1}{2} \right) equals :

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  2. Question 2

    This type asked 19 times: 3 in 2021, 6 in 2022, 4 in 2023, 3 in 2024, 1 in 2025, 2 in 2026. This one: 2 Apr 2026 S2 See all 19 of this type

    70
    Let x=x(y)x = x(y) be the solution of the differential equation 2y2dxdy−2xy+x2=0,y>1,x(e)=e2y^{2}\frac{dx}{dy}- 2xy+x^{2}= 0,y> 1,x(e) =e. Then x(e2)x\left( e^{2} \right) is equal to:

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  3. Question 3

    This type asked 13 times: 2021 Paper 7, 2021 Paper 11, 2021 Paper 16, 2021 Paper 17, 29 Jan 2023, 27 Jan 2024, 30 Jan 2024, 4 Apr 2024, 6 Apr 2024, 9 Apr 2024, 23 Jan 2025, 22 Jan 2026 S2, 2 Apr 2026 S1. This one: 2 Apr 2026 S1 See all 13 of this type

    70
    Let y=y(x)y=y(x) be the solution curve of the differential equation (1+sin⁡x)dydx+(y+1)cos⁡x=0,y(0)=0(1 + \sin x)\frac{dy}{dx}+ (y+ 1)\cos x= 0,y(0) = 0. If the curve y=y(x)y = y(x) passes through the point (α,−12)\left( \alpha,\frac{- 1}{2} \right), then a value of α\alpha is :

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  4. Question 4

    This type asked 8 times: 2021 Paper 26, 31 Jan 2023, 3 Apr 2025, 4 Apr 2025, 22 Jan 2026 S1, 23 Jan 2026 S1, 5 Apr 2026 S2, 6 Apr 2026 S2. This one: 6 Apr 2026 S2 See all 8 of this type

    68
    Let f:R→Rf:R\rightarrow R be such that f(xy)=f(x)f(y)f(xy) =f(x)f(y), for all x,y∈Rx,y\in R and f(0)≠0f(0) \neq 0. Let g:[1,∞)→Rg:\lbrack 1,\infty) \rightarrow R be a differentiable function such that x2g(x)=∫1x(t2f(t)−tg(t))dtx^{2}g(x) =\int_{1}^{x} \left( t^{2}f(t) - tg(t) \right)dtThen g(2)g(2) is equal to :

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  5. Question 5

    This type asked 7 times: 2021 Paper 19, 2021 Paper 25, 6 Apr 2023, 31 Jan 2024, 8 Apr 2024, 4 Apr 2025, 28 Jan 2026 S1. This one: 28 Jan 2026 S1 See all 7 of this type

    70
    Let y=y(x)y=y(x) be the solution of the differential equation xdydx−sin⁡2y=x3(2−x3)cos⁡2y,x≠0.x\frac{dy}{dx}- \sin{2y}=x^{3}\left( 2 -x^{3} \right)\cos^{2}y,x\neq 0. If y(2)=0y(2) =0, then tan⁡(y(1))\tan(y(1)) is equal to

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  6. Question 6

    This type asked 7 times: 2021 Paper 2, 27 Jun 2022, 28 Jun 2022, 26 Jul 2022, 5 Apr 2024, 22 Jan 2025, 7 Apr 2025. This one: 7 Apr 2025 See all 7 of this type

    138
    Let y=y(x)y=y\left( x \right) be the solution of the differential equation (x2+1)y′−\left( x^{2}+ 1 \right)y^{'}- 2xy=(x4+2x2+1)cos⁡x2xy=\left( x^{4}+ 2x^{2}+ 1 \right)\cos x, y(0)=1y(0) = 1. Then ∫−33y(x)dx\int_{- 3}^{3} y(x)dx is :

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  7. Question 7

    This type asked 6 times: 2021 Paper 21, 14 Jun 2022, 27 Jun 2022, 8 Apr 2023, 15 Apr 2023, 22 Jan 2025. This one: 22 Jan 2025 See all 6 of this type

    135
    If x=f(y)x = f(y) is the solution of the differential equation
    (1+y2)+(x−2etan⁡−1y)dydx=0,y∈(−π2,π2)\left( 1 +y^{2} \right)+\left( x - 2e^{\tan^{- 1}y} \right)\frac{dy}{dx}= 0,y \in\left( -\frac{\pi}{2},\frac{\pi}{2} \right)
    with f(0)=1f(0) = 1, then f(13)f\left( \frac{1}{\sqrt{3}} \right) is equal to :

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  8. Question 8

    This type asked 6 times: 2021 Paper 18, 26 Jul 2022, 29 Jul 2022, 25 Jan 2023, 1 Feb 2024, 9 Apr 2024. This one: 9 Apr 2024 See all 6 of this type

    173
    For a differentiable function f:IR→IRf:IR \rightarrow IR, suppose f′(x)=3f(x)+αf^{'}(x) = 3f(x) + \alpha, where α∈IR,f(0)=1\alpha \in IR,f(0) = 1 and lim⁡x→−∞f(x)=7\lim_{x \rightarrow - \infty} f(x) = 7. Then 9f(−log⁡e3)9f\left( -\log_{e}3 \right) is equal to

  9. Question 9

    This type asked 5 times: 2021 Paper 5, 25 Jun 2022, 26 Jun 2022, 26 Jul 2022, 23 Jan 2026 S2. This one: 23 Jan 2026 S2 See all 5 of this type

    71
    If the solution curve y=f(x)y = f(x) of the differential equation (x2−4)y′−2xy+2x(4−x2)2=0, x>2\left( x^{2} - 4 \right)y' - 2xy + 2x\left( 4 - x^{2} \right)^{2} = 0,\ x > 2, passes through the point ( 3,15 ), then the local maximum value of f is ____\_\_\_\_

  10. Question 10

    This type asked 5 times: 2021 Paper 9, 2021 Paper 21, 2021 Paper 25, 28 Jul 2022, 5 Apr 2024. This one: 5 Apr 2024 See all 5 of this type

    152
    The differential equation of the family of circles passing the origin and having center at the line y=xy = x is:

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