Linear and homogeneous

Separable and not linear

Both separable and linear

Neither separable nor linear

Linear

Exact

Homogeneous

Separable

$$\ln\left(x^2\right)$$  

$$-\frac{1}{x^2}$$  

$$-\frac{1}{\sqrt{x}}$$  

$$\frac{1}{\sqrt{x}}$$  

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

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

$$\frac{\text{d}x}{\text{d}y}+\left(2x-1\right)y=0$$

$$\frac{\partial N}{\partial x}=xe^y+1$$  

$$\frac{\partial M}{\partial y}=ye^y$$  

$$\frac{\partial M}{\partial y}=e^y$$  

$$k=2$$  

$$k=10$$  

$$k=-12$$  

$$k=-10$$  

$\frac{\text{d}x}{\text{d}y}-\frac{2}{x}=e^x$ is linear and its integrating factor is $-\frac{1}{x^2}.$

$\frac{\text{d}x}{\text{d}y}-4x=0\$ with condition $y\left(1\right)=2$ is linear and its solution is $y=2x^2.$

$\frac{\text{d}x}{\text{d}y}-\frac{2}{x}=e^x\$ is linear and its solution is $y=\frac{1}{x^2}.$

$y^3dx-4x^2dy=0$ is not exact.

Linear and not separable

Exact

Homogeneous and linear

Linear and separable

 $y=-20+6e^{-2x}+Ce^{4x}$  

 $y=-\frac{5}{4}e^{-4x}+\frac{1}{6}e^{-2x}+Ce^{4x}$  

 $y=-20e^{-8x}+e^{-10x}+Ce^{-4x}$  

 $y=-\frac{5}{4}+\frac{1}{6}e^{-2x}+Ce^{-4x}$