Solving Linear Differential Equations of First Order

Q. The family of curves satisfying the differential equation $\frac{dy}{dx}+\frac{1}{x}\sin 2y=x^3{\cos }^{2}y,$ is
(where $C$ is an arbitrary constant)

1. $x^6+6x^2=C\tan y$
2. $6x^2\tan y=x^6+C$
3. $\sin 2y=x^3{\cos }^{2}y+C$
4. $y^6=6y^2\tan x+C$

Q. The solution of $y_2-7y_1+12y=0$ is

1. $y=C_1{e}^{3x}+C_2{e}^{4x}$
2. $y=C_{1}x{e}^{3x}+C_2{e}^{4x}$
3. $y=C_1{e}^{3x}+C_{2}x{e}^{4x}$
4. $Noneofthese$

Q. If $y=y\left(x\right)$ is the solution of the equation $e^{\sin y}\cos y\frac{dy}{dx}+e^{\sin y}\cos x=\cos x,$ $y\left(0\right)=0;$ then $1+y\left(\frac{\pi }{6}\right)+\frac{\sqrt{3}}{2}y\left(\frac{\pi }{3}\right)+\frac{1}{\sqrt{2}}y\left(\frac{\pi }{4}\right)$ is equal to

Q. Let $y=f\left(x\right)$ be a real-valued differentiable function on the set of all real numbers $R$ such that $f\left(1\right)=1$. If $f\left(x\right)$ satisfies $x{f}^{\prime }\left(x\right)=x^2+f\left(x\right)-2$, then

1. $f\left(x\right)$ is even function
2. $f\left(x\right)$ is odd function
3. minimum value of $f\left(x\right)$ is $0$
4. $y=f\left(x\right)$ represent a parabola with focus $(1,\frac{5}{4})$

Q. If $y=y\left(x\right)$ is the solution of the differential equation, $x\frac{dy}{dx}+2y=x^2$ satisfying $y\left(1\right)=1$, then $y\left(\frac{1}{2}\right)$ is equal to:

1. $\frac{1}{4}$
2. $\frac{7}{64}$
3. $\frac{13}{16}$
4. $\frac{49}{16}$

Q. The solution of the given differential equation $\frac{dy}{dx}+2xy=y$ is

1. $y=c{e}^{x-x^2}$
   
2. $y=c{e}^{x^2-x}$
   
3. $y=c{e}^x$
   
4. $y=c{e}^{-x^2}$

Q. The solution of $\frac{dv}{dt}+\frac{k}{m}v=-g$ is

1. $v=c{e}^{-\frac{k}{m}t}-\frac{mg}{k}$
2. $v=c-\frac{mg}{k}e-\frac{k}{m}t$
3. $ve-\frac{k}{m}t=c-\frac{mg}{k}$
4. $ve\frac{k}{m}t=c-\frac{mg}{k}$

Q. If $y\left(x\right)$ is the solution of the differential equation
$(x+2)\frac{dy}{dx}=x^2+4x–9,x\ne –2$
$\text{and}y\left(0\right)=0,\text{then}y(–4)$is equal to :

1. $2$
2. $0$
3. $-1$
4. $1$

Q. The curves satisfying the differential equation $(1-x^2)y^1+xy=ax$ are

1. ellipses and hyperbolas
2. ellipses and parabola
3. ellipses and straight lines
4. circles and ellipses

Q. If $y_1\left(x\right)$ is a solution of the differential equation $\frac{dy}{dx}+f\left(x\right)y=0$, then a solution of differential equation $\frac{dy}{dx}+f\left(x\right)y=r\left(x\right)$ is

1. $\frac{1}{y\left(x\right)}\int y_1\left(x\right)dx$
2. $y_1\left(x\right)\int \frac{r\left(x\right)}{y_1\left(x\right)}dx+c$
3. $intr\left(x\right)y_1\left(x\right)dx$
4. $noneofthese$

Q. Consider $I_1=\underset{0}{\overset{\pi /4}{\int }}{e}^{x^2}dx,I_2=\underset{0}{\overset{\pi /4}{\int }}{e}^{x}dx,$
$I_3=\underset{0}{\overset{\pi /4}{\int }}{e}^{x^2}\cos xdx,I_4=\underset{0}{\overset{\pi /4}{\int }}{e}^{x^2}\sin xdx$
Statement $1:I_2>I_1>I_3>I_4$
Statement $2:$ For $x\in (0,\frac{\pi }{4}),x>x^2$ and $\cos x>\sin x$

1. Both statements $1$ and $2$ are true but statement $2$ is not the correct explanation of statement $1.$
2. Statement $1$ is true but statement $2$ is false.
3. Both statements $1$ and $2$ are true and statement $2$ is the correct explanation of statement $1.$
4. Statement $1$ is false but statement $2$ is true.

Q. Let $y=y\left(x\right)$ be the solution of the differential equation, $(x^2+1{)}^2\frac{dy}{dx}+2x(x^2+1)y=1$ such that $y\left(0\right)=0$. If $\sqrt{a}y\left(1\right)=\frac{\pi }{32}$, then the value of $a$ is:

1. $\frac{1}{16}$
2. $\frac{1}{2}$
3. $1$
4. $\frac{1}{4}$