Differential Equations · JEE Main previous year questions

Let $y(x)$ be the solution of the differential equation $(x\log x)\frac{dy}{dx}+y=2x\log x,(x\ge 1)$. Then $y(e)$ is equal to :

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

Let the population of rabbits surviving at time $t$ be governed by the differential equation $\frac{dp(t)}{dt}=\frac{1}{2}p(t)-200$. If $p(0)=100$, then $p(t)$ equals:

1. A.$600-500e^{t/2}$
2. B.$400-300e^{-t/2}$
3. C.$400-300e^{t/2}$
4. D.$300-200e^{-t/2}$

The integrating factor of the differential equation $(x^2-1)\frac{dy}{dx}+2xy=x$ is

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

The differential equation which represents the family of curves $y=c_1{e}^{c_{2}x}$, where $c_1$, and $c_2$ are arbitrary constants, is

1. A.$y^{\prime \prime }=y^{\prime }y$
2. B.$y{y}^{\prime \prime }=y^{\prime }$
3. C.$y{y}^{\prime \prime }=(y^{\prime }{)}^2$
4. D.$y^{\prime }=y^2$

The differential equation whose solution is $A{x}^2+B{y}^2=1$ where $A$ and $B$ are arbitrary constants is of

1. A.second order and second degree
2. B.first order and second degree
3. C.first order and first degree
4. D.second order and first degree

The differential equation representing the family of curves $y^2=2c\left(x+\sqrt{c}\right)$, where $c>0$, is a parameter, is of order and degree as follows:

1. A.order 1, degree 2
2. B.order 1, degree 1
3. C.order 1, degree 3
4. D.order 2, degree 2

If $x\frac{dy}{dx}=y(\log y-\log x+1)$, then the solution of the equation is :

1. A.$y\log \left(\frac{x}{y}\right)=cx$
2. B.$x\log \left(\frac{y}{x}\right)=cy$
3. C.$\log \left(\frac{y}{x}\right)=cx$
4. D.$\log \left(\frac{x}{y}\right)=cy$

Solution of the differential equation $ydx+(x+x^{2}y)dy=0$ is

1. A.$\log y=Cx$
2. B.$-\frac{1}{xy}+\log y=C$
3. C.$\frac{1}{xy}+\log y=C$
4. D.$-\frac{1}{xy}=C$

The differential equation for the family of circle $x^2+y^2-2ay=0$, where a is an arbitrary constant is :

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

The order and degree of the differential equation ${\left(1+3\frac{dy}{dx}\right)}^{2/3}=4\frac{d^{3}y}{d{x}^3}$ are

1. A.$\left(1,\frac{2}{3}\right)$
2. B.$(3,1)$
3. C.$(3,3)$
4. D.$(1,2)$

The solution of the equation $\frac{d^{2}y}{d{x}^2}=e^{-2x}$

1. A.$\frac{e^{-2x}}{4}$
2. B.$\frac{e^{-2x}}{4}+cx+d$
3. C.$\frac{1}{4}{e}^{-2x}+c{x}^2+d$
4. D.$\frac{1}{4}{e}^{-4x}+cx+d$