Differential Equations · JEE Main previous year questions

2025 8 questions

1. Q22025 · 8 Apr 2025 · Shift 2

   Let $f(x)=x-1$ and $g(x)=e^x$ for $x\in \mathbb{R}$. If $\frac{dy}{dx}=\left(e^{-2\sqrt{x}}g\left(f(f(x))\right)-\frac{y}{\sqrt{x}}\right)$, $y(0)=0$, then $y(1)$ is

   1. A.$\frac{1-e^3}{e^4}$
   2. B.$\frac{e-1}{e^4}$
   3. C.$\frac{1-e^2}{e^4}$
   4. D.$\frac{2e-1}{e^3}$
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   Answer: B

2. Q72025 · 4 Apr 2025 · Shift 2

   If a curve $y=y(x)$ passes through the point $\left(1,\frac{\pi }{2}\right)$ and satisfies the differential equation $\left(7x^4\cot y-e^x\mathrm{cosec}y\right)\frac{dx}{dy}=x^5,x\ge 1$, then at $x=2$, the value of $\cos y$ is :

   1. A.$\frac{2e^2+e}{64}$
   2. B.$\frac{2e^2-e}{64}$
   3. C.$\frac{2e^2-e}{128}$
   4. D.$\frac{2e^2+e}{128}$
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   Answer: C

3. Q142025 · 3 Apr 2025 · Shift 1

   Let $g$ be a differentiable function such that ${\int }_0^{x}g(t)dt=x-{\int }_0^{x}tg(t)dt$, $x\ge 0$ and let $y=y(x)$ satisfy the differential equation $\frac{dy}{dx}-y\tan x=2(x+1)\sec xg(x)$, $x\in \left[0,\frac{\pi }{2}\right)$. If $y(0)=0$, then $y\left(\frac{\pi }{3}\right)$ is equal to

   1. A.$\frac{4\pi }{3}$
   2. B.$\frac{2\pi }{3}$
   3. C.$\frac{2\pi }{3\sqrt{3}}$
   4. D.$\frac{4\pi }{3\sqrt{3}}$
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   Answer: A

4. Q82025 · 3 Apr 2025 · Shift 2

   Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}+3\left({\tan }^{2}x\right)y+3y={\sec }^{2}x,y(0)=\frac{1}{3}+e^3$. Then $y\left(\frac{\pi }{4}\right)$ is equal to :

   1. A.$\frac{4}{3}$
   2. B.$\frac{2}{3}+e^3$
   3. C.$\frac{4}{3}+e^3$
   4. D.$\frac{2}{3}$
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   Answer: A

5. Q242025 · 2 Apr 2025 · Shift 2· Numerical answer

   Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}+2y{\sec }^{2}x=2{\sec }^{2}x+3\tan x\cdot {\sec }^{2}x$ such that $y(0)=\frac{5}{4}$. Then $12\left(y\left(\frac{\pi }{4}\right)-e^{-2}\right)$ is equal to ____.

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   Answer: 21

6. Q132025 · 29 Jan 2025 · Shift 2

   If for the solution curve $y=f(x)$ of the differential equation $\frac{dy}{dx}+(\tan x)y=\frac{2+\sec x}{(1+2\sec x{)}^2}$ , $x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$, $f\left(\frac{\pi }{3}\right)=\frac{\sqrt{3}}{10}$, then $f\left(\frac{\pi }{4}\right)$ is equal to:

   1. A.$\frac{5-\sqrt{3}}{2\sqrt{2}}$
   2. B.$\frac{4-\sqrt{2}}{14}$
   3. C.$\frac{9\sqrt{3}+3}{10(4+\sqrt{3})}$
   4. D.$-\frac{\sqrt{3}+1}{10(4+\sqrt{3})}$
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   Answer: B

7. Q232025 · 24 Jan 2025 · Shift 2· Numerical answer

   Let $y=y(x)$ be the solution of the differential equation $2\cos x\frac{dy}{dx}=\sin 2x-4y\sin x,x\in \left(0,\frac{\pi }{2}\right)$. If $y\left(\frac{\pi }{3}\right)=0$, then $y^{\prime }\left(\frac{\pi }{4}\right)+y\left(\frac{\pi }{4}\right)$ is equal to ________.

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   Answer: 1

8. Q202025 · 23 Jan 2025 · Shift 2

   Let $x=x(y)$ be the solution of the differential equation $y=\left(x-y\frac{\mathrm{d}x}{\mathrm{d}y}\right)\sin \left(\frac{x}{y}\right),y>0$ and $x(1)=\frac{\pi }{2}$. Then $\cos (x(2))$ is equal to :

   1. A.$2({\log }_{\mathrm{e}}2)-1$
   2. B.$1-2({\log }_{\mathrm{e}}2{)}^2$
   3. C.$1-2({\log }_{\mathrm{e}}2)$
   4. D.$2({\log }_{\mathrm{e}}2{)}^2-1$
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   Answer: D