First Order Linear Differential Equations

Which of the following differential equations is not linear?

•  $\frac{\text{d}y}{\text{d}x}+xy^2=e^x$  

•  $\frac{1}{x}\frac{\text{d}y}{\text{d}x}-y=x+1$  

•  $x\frac{\text{d}y}{\text{d}x}+y=x-x^2y$  

•  $\frac{\text{d}y}{\text{d}x}=1-xy$  

The general solution of the differential equation

 $\frac{\text{d}y}{\text{d}x}=2xy$ 

 is

•  $y=e^{x^2}+A$  

•  $y=Ae^{x^2}$  

•  $y=e^{x^2}$  

•  $y=\ln\left(x^2+A\right)$  

The particular solution of the differential equation

 $\frac{\text{d}y}{\text{d}x}=y^2e^{2x}$ 

for which y=1 when x=0 is given by

 

•  $y=\frac{2}{3-e^{2x}}$  

•  $y=\frac{2}{1+e^{2x}}$  

•  $y=\frac{1}{2-e^{2x}}$  

•  $y=e^{-2x}$  

An integrating factor of the differential equation

 $\frac{\text{d}y}{\text{d}x}-\frac{2y}{x}=3x^2$ 

is
 

•  $e^{-2x}$  

•  $-\ln\left(x^2\right)$  

•  $x^2$  

•  $\frac{1}{x^2}$  

An integrating factor of the differential equation

 $\left(e^{2x}+1\right)\frac{\text{d}y}{\text{d}x}+4e^{2x}y=x$ 

is
 

•  $e^{2x}+1$  

•  $e^{2e^{2x}}$  

•  $\left(e^{2x}+1\right)^2$  

•  $\left(e^{2x}+1\right)^4$  

The general solution of the differential equation

 $\frac{\text{d}y}{\text{d}x}-\left(\cot x\right)y=\sin\left(2x\right)$ 

 is

•  $y=\frac{1}{2}\sin\left(2x\right)+A\sin x$  

•  $y=2\sin^2\left(x\right)+\sin x+A$  

•  $y=\frac{2}{3}\sin^2\left(x\right)+A\operatorname{cosec}\left(x\right)$  

•  $y=2\sin^2\left(x\right)+A\sin x$  

The general solution of the differential equation

 $x\frac{\text{d}y}{\text{d}x}+3y=x^2$ 

 is

•  $y=\frac{x^2}{5}+\frac{c}{x^3}$  

•  $y=x^2+Cx$  

•  $y=\frac{x}{5}+C$  

•  $y=\frac{x^2}{2}+\frac{c}{x}$  

The particular solution of the differential equation

 $\frac{\text{d}y}{\text{d}x}+2y=x$ 

for which y=0 when x=0 is given by

 

•  $y=\frac{1}{2}x^2e^{-2x}$  

•  $y=\frac{e^{-2x}+2x-1}{4}$  

•  $y=1-e^{-2x}$  

•  $y=\frac{1-e^{-2x}}{2}$  

The particular solution of the differential equation

 $\left(x^2+1\right)\frac{\text{d}y}{\text{d}x}+xy=x\left(x^2+1\right)$ 

for which y=0 when x=0 is given by

 

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

•  $y=\frac{x^2\left(x^2+2\right)}{4\left(x^2+1\right)}$  

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

•  $y=\frac{1}{3}\left(x^2+1\right)+\frac{1}{3}\left(x^2+1\right)^{\frac{1}{2}}$  

The general solution of the differential equation

 $\frac{\text{d}y}{\text{d}x}+y=2e^{-x}$ 

 is

•  $y=e^{-x}\left(A+2x\right)$  

•  $y=Ae^{-x}+2e^{-x}$  

•  $y=Ae^x+2xe^{-x}$  

•  $y=Ae^{-x}+Bxe^{-x}$