Bernoulli's Equation

Q.

The solution of the differential equation $\frac{dy}{dx}-\frac{y+3x}{\ln \left(y+3x\right)}+3=0$ is: (where $C$ is a constant of integration)

1. $x-\ln \left(y+3x\right)=C$

2. $x-\frac{1}{2}\ln {\left(y+3x\right)}^2=C$

3. $x-2\ln \left(y+3x\right)=C$

4. $y+3x-\frac{1}{2}{\left(\ln \left(x\right)\right)}^2=C$

Q. The general solution of $\frac{dy}{dx}-\frac{y}{x}=\frac{y^2}{x^2}$ is

1. $\frac{x}{y}=-ln{x}^2+C$
2. $\frac{x}{y}=-ln\frac{x}{2}+C$
3. $\frac{x}{y}=-lnx+C$
4. $Noneofthese$

Q. If $\frac{dy}{dx}-y=y^2(\sin x+\cos x)$ with $y\left(0\right)=1,$ then the value of $y\left(\pi \right)$ is

1. $e^{\pi }$
2. $-e^{\pi }$
3. $e^{-\pi }$
4. $-e^{-\pi }$

Q. The general solution of $\frac{dy}{dx}-\frac{y}{x}=\frac{y^2}{x^2}$ is

1. $\frac{x}{y}=-ln{x}^2+C$
2. $\frac{x}{y}=-ln\frac{x}{2}+C$
3. $\frac{x}{y}=-lnx+C$
4. $Noneofthese$

Q. A function $y=f\left(x\right)$ satisfies $x{f}^{\prime }\left(x\right)-2f\left(x\right)=x^4\left(f\right(x){)}^2$ for all $x>0$ and $f\left(1\right)=-6.$ Then the value of $f\left(2\right)$ is

1. $-\frac{2}{3}$
2. $-\frac{3}{8}$
3. $\frac{2}{3}$
4. $\frac{1}{2}$

Q. A function $y=f\left(x\right)$ satisfies $x{f}^{\prime }\left(x\right)-2f\left(x\right)=x^4{f}^2\left(x\right),\forall x>0$ and $f\left(1\right)=-6.$ Then the value of $f^{\prime }\left(3^{1/5}\right)$ is

Q.

The integrating factor of the differential equation $\frac{dy}{dx}=ytanx-y^{2}secx$ is

[MP PET 1995; Pb. CET 2002]

1. tan x

2. secx

3. -sec x

4. cot x

Q. $\mathrm{Which}\mathrm{of}\mathrm{the}\mathrm{following}\mathrm{is}\mathrm{the}\mathrm{solution}\mathrm{of}\mathrm{the}\mathrm{differential}\mathrm{equation}x\frac{\mathrm{dy}}{\mathrm{dx}}+y={\mathrm{xy}}^3?$

1. $y^2=\frac{1}{2x+{\mathrm{cx}}^2}$
2. $y^{}=\frac{1}{2x+{\mathrm{cx}}^2}$
3. $y^2=2x+{\mathrm{cx}}^2$
4. $y^3=\frac{1}{2x+{\mathrm{cx}}^2}$