Solving Linear Differential Equations of First Order

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

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

Q. The solution of $(x+2y^3)\left(\frac{dy}{dx}\right)=y$ is (where c is arbitrary constant)

1. $x=y^3+cy$
2. $x=y^3-cy$
3. $y=y^3-cy$
4. $y=y^3+cy$

Q. Solution of the differential equation
$cosxdy=y(sinx-y)dx,0<x<\frac{\pi }{2}$ is

1. $ysecx=tanx+c$
2. $ytanx=secx+c$
3. $tanx=(secx+c)y$
4. $secx=(tanx+c)y$

Q.

The solution of the differential equation $x\frac{dy}{dx}+y=x^2+3x+2$ is

1. $xy=\frac{x^3}{3}+\frac{3}{2}{x}^2+2x+c$

2. $xy=\frac{x^4}{4}+x^3+x^2+c$

3. $xy=\frac{x^4}{4}+\frac{3}{3}{x}^2+c$

4. $xy=\frac{x^4}{4}+x^3+x^2+cx$

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.

The solution of the equation $\frac{dy}{dx}+ytanx=x^{m}cosx$ is

1. $(m+1)y=x^{m+1}cosx+c(m+1)cosx$
2. $(my=(x^m+c)cosx$
3. $y=(x^{m+1}+c)cosx$

4. $Noneofthese$

Q.

Let $f:R^{+}\to R^{+}$ be a differentiable function satisfying $f\left(xy\right)=\frac{f\left(x\right)}{y}+\frac{f\left(y\right)}{x}$ for all $x,yϵR^{+}$. Also $f\left(1\right)=0,f^{\prime }\left(1\right)=1$. If M is the greatest value of $f\left(x\right)$ then $[m+e]$ is ___ (where [.] represents Greatest Integer Function).

1. 3

2. 2

3. 1

4. 0

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.

The solution of the differential equation $\frac{dy}{dx}+\frac{3x^2}{1+x^3}y=\frac{si{n}^{2}x}{1+x^3}$ is

1. $y(1+x^3)=x+\frac{1}{2}sin2x+c$

2. $2ta{n}^{-1}\left(\frac{x}{y}\right)+logx+c=0$

3. $log(y+\sqrt{x^2+y^2})+logy+c=0$

4. $sin{h}^{-1}\left(\frac{x}{y}\right)+logy+c=0$