Differential Equations · JEE Main previous year questions

If $y=y(x)$ is the solution curve of the differential equation $$\frac{dy}{dx}+y\tan x=x\sec x,0\le x\le \frac{\pi }{3},y(0)=1$$ then $y\left(\frac{\pi }{6}\right)$ is equal to

1. A.$\frac{\pi }{12}-\frac{\sqrt{3}}{2}{\log }_e\left(\frac{2\sqrt{3}}{e}\right)$
2. B.$\frac{\pi }{12}+\frac{\sqrt{3}}{2}{\log }_e\left(\frac{2\sqrt{3}}{e}\right)$
3. C.$\frac{\pi }{12}+\frac{\sqrt{3}}{2}{\log }_e\left(\frac{2}{e\sqrt{3}}\right)$
4. D.$\frac{\pi }{12}-\frac{\sqrt{3}}{2}{\log }_e\left(\frac{2}{e\sqrt{3}}\right)$

Let the solution curve $y=y(x)$ of the differential equation $$\frac{dy}{dx}-\frac{3x^5{\tan }^{-1}(x^3)}{(1+x^6{)}^{3/2}}y=2x\exp \left\{\frac{x^3-{\tan }^{-1}{x}^3}{\sqrt{(1+x^6)}}\right\}$$ pass through the origin. Then $y(1)$ is equal to :

1. A.$\exp \left(\frac{1-\pi }{4\sqrt{2}}\right)$
2. B.$\exp \left(\frac{4-\pi }{4\sqrt{2}}\right)$
3. C.$\exp \left(\frac{4+\pi }{4\sqrt{2}}\right)$
4. D.$\exp \left(\frac{\pi -4}{4\sqrt{2}}\right)$

Let $y=y(x)$ be the solution of the differential equation $x{\log }_{e}x\frac{dy}{dx}+y=x^2{\log }_{e}x,(x>1)$. If $y(2)=2$, then $y(e)$ is equal to

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

Let $y=y(x)$ be the solution curve of the differential equation $\frac{dy}{dx}=\frac{y}{x}\left(1+x{y}^2(1+{\log }_{e}x)\right)$, $x>0,y(1)=3$. Then $\frac{y^2(x)}{9}$ is equal to :

1. A.$\frac{x^2}{5-2x^3(2+{\log }_e{x}^3)}$
2. B.$\frac{x^2}{3x^3(1+{\log }_e{x}^2)-2}$
3. C.$\frac{x^2}{7-3x^3(2+{\log }_e{x}^2)}$
4. D.$\frac{x^2}{2x^3(2+{\log }_e{x}^3)-3}$

Let the solution curve $y=y(x)$ of the differential equation $\left(1+{\mathrm{e}}^{2x}\right)\left(\frac{\mathrm{d}y}{\mathrm{d}x}+y\right)=1$ pass through the point $\left(0,\frac{\pi }{2}\right)$. Then, ${\lim }_{x\to \infty }{\mathrm{e}}^{x}y(x)$ is equal to :

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

If the solution curve of the differential equation $\frac{dy}{dx}=\frac{x+y-2}{x-y}$ passes through the points $(2,1)$ and $(k+1,2)$, $k>0$, then

1. A.$2{\tan }^{-1}\left(\frac{1}{k}\right)={\log }_e\left(k^2+1\right)$
2. B.${\tan }^{-1}\left(\frac{1}{k}\right)={\log }_e\left(k^2+1\right)$
3. C.$2{\tan }^{-1}\left(\frac{1}{k+1}\right)={\log }_e\left(k^2+2k+2\right)$
4. D.$2{\tan }^{-1}\left(\frac{1}{k}\right)={\log }_e\left(\frac{k^2+1}{k^2}\right)$

Let $y=y(x)$ be the solution curve of the differential equation $\frac{dy}{dx}+\left(\frac{2x^2+11x+13}{x^3+6x^2+11x+6}\right)y=\frac{(x+3)}{x+1}$, $x>-1$, which passes through the point $(0,1)$. Then $y(1)$ is equal to :

1. A.$\frac{1}{2}$
2. B.$\frac{3}{2}$
3. C.$\frac{5}{2}$
4. D.$\frac{7}{2}$

Let the solution curve of the differential equation $xdy=\left(\sqrt{x^2+y^2}+y\right)dx$, $x>0$, intersect the line $x=1$ at $y=0$ and the line $x=2$ at $y=\alpha$. Then the value of $\alpha$ is :

1. A.$\frac{1}{2}$
2. B.$\frac{3}{2}$
3. C.$-\frac{3}{2}$
4. D.$\frac{5}{2}$

If $y=y(x),x\in (0,\pi /2)$ be the solution curve of the differential equation $$({\sin }^{2}2x)\frac{dy}{dx}+(8{\sin }^{2}2x+2\sin 4x)y=2{\mathrm{e}}^{-4x}(2\sin 2x+\cos 2x),$$ with $y(\pi /4)={\mathrm{e}}^{-\pi }$, then $y(\pi /6)$ is equal to :

1. A.$\frac{2}{\sqrt{3}}{\mathrm{e}}^{-2\pi /3}$
2. B.$\frac{2}{\sqrt{3}}{\mathrm{e}}^{2\pi /3}$
3. C.$\frac{1}{\sqrt{3}}{\mathrm{e}}^{-2\pi /3}$
4. D.$\frac{1}{\sqrt{3}}{\mathrm{e}}^{2\pi /3}$

Let $A=\left[\begin{array}{cc}1& -1\\ 2& \alpha \end{array}\right]$ and $B=\left[\begin{array}{cc}\beta & 1\\ 1& 0\end{array}\right],\alpha ,\beta \in \mathbf{R}$. Let ${\alpha }_1$ be the value of $\alpha$ which satisfies $(A+B{)}^2=A^2+\left[\begin{array}{cc}2& 2\\ 2& 2\end{array}\right]$ and ${\alpha }_2$ be the value of $\alpha$ which satisfies $(A+B{)}^2=B^2$. Then $|{\alpha }_1-{\alpha }_2|$ is equal to ____.

Let $y=y(x)$ be the solution curve of the differential equation $\frac{dy}{dx}+\frac{1}{x^2-1}y={\left(\frac{x-1}{x+1}\right)}^{1/2}$, $x>1$ passing through the point $\left(2,\sqrt{\frac{1}{3}}\right)$. Then $\sqrt{7}y(8)$ is equal to :

1. A.$11+6{\log }_{e}3$
2. B.19
3. C.$12-2{\log }_{e}3$
4. D.$19-6{\log }_{e}3$

The differential equation of the family of circles passing through the points $(0,2)$ and $(0,-2)$ is :

1. A.$2xy\frac{dy}{dx}+(x^2-y^2+4)=0$
2. B.$2xy\frac{dy}{dx}+(x^2+y^2-4)=0$
3. C.$2xy\frac{dy}{dx}+(y^2-x^2+4)=0$
4. D.$2xy\frac{dy}{dx}-(x^2-y^2+4)=0$

Let $y=y_1(x)$ and $y=y_2(x)$ be two distinct solutions of the differential equation $\frac{dy}{dx}=x+y$, with $y_1(0)=0$ and $y_2(0)=1$ respectively. Then, the number of points of intersection of $y=y_1(x)$ and $y=y_2(x)$ is

1. A.0
2. B.1
3. C.2
4. D.3

Let $y=y(x)$ be the solution curve of the differential equation $\sin \left(2x^2\right){\log }_e\left(\tan x^2\right)dy+\left(4xy-4\sqrt{2}x\sin \left(x^2-\frac{\pi }{4}\right)\right)dx=0$, $0<x<\sqrt{\frac{\pi }{2}}$, which passes through the point $(\sqrt{\frac{\pi }{6}},1)$. Then $\left|y\left(\sqrt{\frac{\pi }{3}}\right)\right|$ is equal to _________.

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=\frac{4y^3+2y{x}^2}{3x{y}^2+x^3}$, $y(1)=1$. If for some $n\in \mathbb{N}$, $y(2)\in [n-1,n)$, then $n$ is equal to ___________.

If the solution curve of the differential equation $\left(\left({\tan }^{-1}y\right)-x\right)dy=\left(1+y^2\right)dx$ passes through the point $(1,0)$, then the abscissa of the point on the curve whose ordinate is $\tan (1)$, is

1. A.$2e$
2. B.$\frac{2}{e}$
3. C.$2$
4. D.$\frac{1}{e}$

Let $y=y(x)$ be the solution of the differential equation $(1-x^2)dy=\left(xy+(x^3+2)\sqrt{1-x^2}\right)dx$, $-1<x<1$, and $y(0)=0$. If ${\int }_{-\frac{1}{2}}^{\frac{1}{2}}\sqrt{1-x^2}y(x)dx=k$, then $k^{-1}$ is equal to ________.

Let the solution curve $y=y(x)$ of the differential equation $$(4+x^2)dy-2x(x^2+3y+4)dx=0$$ pass through the origin. Then $y(2)$ is equal to ____.

Let $S=(0,2\pi )-\left\{\frac{\pi }{2},\frac{3\pi }{4},\frac{3\pi }{2},\frac{7\pi }{4}\right\}$. Let $y=y(x),x\in S$, be the solution curve of the differential equation $\frac{dy}{dx}=\frac{1}{1+\sin 2x},y\left(\frac{\pi }{4}\right)=\frac{1}{2}$. If the sum of abscissas of all the points of intersection of the curve $y=y(x)$ with the curve $y=\sqrt{2}\sin x$ is $\frac{k\pi }{12}$, then $k$ is equal to ____.

If $y=y(x)$ is the solution of the differential equation $x\frac{dy}{dx}+2y=x{e}^x$, $y(1)=0$ then the local maximum value of the function $z(x)=x^{2}y(x)-e^x$, $x\in R$ is :

1. A.$1-e$
2. B.0
3. C.$\frac{1}{2}$
4. D.$\frac{4}{e}-e$

If the solution of the differential equation $\frac{dy}{dx}+e^x(x^2-2)y=(x^2-2x)(x^2-2)e^{2x}$ satisfies $y(0)=0$, then the value of $y(2)$ is ____________.

1. A.$-1$
2. B.1
3. C.0
4. D.e

Let $y=y(x)$ be the solution of the differential equation $(x+1)y^{\prime }-y=e^{3x}(x+1{)}^2$, with $y(0)=\frac{1}{3}$. Then, the point $x=-\frac{4}{3}$ for the curve $y=y(x)$ is :

1. A.not a critical point
2. B.a point of local minima
3. C.a point of local maxima
4. D.a point of inflection

If the solution curve $y=y(x)$ of the differential equation $y^{2}dx+(x^2-xy+y^2)dy=0$, which passes through the point $(1,1)$ and intersects the line $y=\sqrt{3}x$ at the point $(\alpha ,\sqrt{3}\alpha )$, then value of ${\log }_e(\sqrt{3}\alpha )$ is equal to :

1. A.$\frac{\pi }{3}$
2. B.$\frac{\pi }{2}$
3. C.$\frac{\pi }{12}$
4. D.$\frac{\pi }{6}$

If $y=y(x)$ is the solution of the differential equation $2x^2\frac{dy}{dx}-2xy+3y^2=0$ such that $y(e)=\frac{e}{3}$, then $y(1)$ is equal to :

1. A.$\frac{1}{3}$
2. B.$\frac{2}{3}$
3. C.$\frac{3}{2}$
4. D.$3$

If $y=y(x)$ is the solution curve of the differential equation $x^{2}dy+\left(y-\frac{1}{x}\right)dx=0$ ; $x>0$ and $y(1)=1$, then $y\left(\frac{1}{2}\right)$ is equal to :

1. A.$\frac{3}{2}-\frac{1}{\sqrt{e}}$
2. B.$3+\frac{1}{\sqrt{e}}$
3. C.$3+e$
4. D.$3-e$