Separation of variables

The differential equation $\frac{\ dy\ }{dx}=xy+2y$  written correctly in separable form is 

•  $\frac{dy}{y+2y}=xdx$  

•  $ydy-y=2xdx$  

•  $\frac{dy}{y}=\left(x+2\right)dx$  

•  $\left(y-2\right)dy=xdx$  

A function  $y=f\left(x\right)$ that satisfies the equation  $y\frac{\text{d}y}{\text{d}x}=x$  and  $f\left(0\right)=1$  is  

•  $f\left(x\right)=x$  

•  $f\left(x\right)=e^x$  

•  $f\left(x\right)=\sqrt{x^2+1}$  

•  $f\left(x\right)=\sqrt{1-x^2}$  

 $\frac{\text{d}x}{\text{d}y}-4y-5+e^{-2x}=0$  

The general solution for the above DE is,

•  $y=-20+6e^{-2x}+Ce^{4x}$  

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

•  $y=-20e^{-8x}+e^{-10x}+Ce^{-4x}$  

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

Which of the following mathematical models CANNOT be solved by using separation of variables

• $\frac{\text{d}P}{\text{dt}}=kP$

• $\frac{dT}{dt}=k\left(T-Tm\right)$

• $\frac{dA}{dt}=kA$

• $L\ \frac{dI}{dt}+RI=E\left(t\right)$

By separation of variables, solve the resulting equations $\int\ \frac{v}{v+1}dv=\int_{ }^{ }\ \frac{1}{y}dy\text{}$  

•  $\frac{x}{y}-\ln\left|\frac{x}{y}+1\right|=\ln y+C$  

•  $\frac{x}{y}+\ln\left|\frac{x}{y}+1\right|=\ln y+C$  

•  $\frac{x}{y}+\frac{x^2}{2y^2}=\ln y+C$  

•  $\frac{x}{y}-\frac{x^2}{2y^2}=\ln y+C$  

 Solve the given differential equations by separable variable method   $\frac{\text{d}v}{\text{d}t}=\frac{3+v^2}{v}$  

•  $v=\ln\left|3+v^2\right|+C$  

•  $v^2=e^{2t+2c}-3$  

•  $v=Ae^{2t}+3,\ \ \ \ \ A=e^{2c}$  

•  $v^2=Ae^{2t}+3,\ \ \ \ \ \ A=C$  

 Solve the given differential equations by separable variable method   $\frac{\text{d}y}{\text{d}x}=e^{-y}\left(2x-4\right),\ \ \ \ \ y\left(5\right)=0$  

•  $y=\ln\left|x^2-4x-4\right|$  

•  $y=x^2-4x-4$  

•  $y=4e^{2t}+4\ \ \ \ \ A=e^{2c}$  

•  $y^2=4e^{2x}+2,\ \ \ \ \ \ A=C$  

The following Differential Equation is
 $\frac{\text{d}y}{\text{d}x}=\frac{y}{x}$  

• Separable.

• Non Separable

The following Differential Equation is
 $\frac{\text{d}y}{\text{d}x}=x^2y+x$  

• Separable.

• Non Separable.

The following Differential Equation is
 $x\frac{\text{d}y}{\text{d}x}=yx^2$  

• Separable.

• Non Separable.