91Ó°ÊÓ

It is sometimes possible to transform a nonexact differential equation \(M(x, y) d x+N(x, y) d y=0\) into an exact equation by multiplying it by an integrating factor \(\mu(x, y)\). In Problems \(37-42\) solve the given equation by verifying that the indicated function \(\mu(x, y)\) is an integrating factor. $$ \left(2 y^{2}+3 x\right) d x+2 x y d y=0, \quad \mu(x, y)=x $$

Solve the given differential equation subject to the indicated initial condition. $$ \left(e^{-7}+1\right) \sin x d x=(1+\cos x) d y, \quad y(0)=0 $$

In Problems \(41-50\) solve the given differential equation subject to the indicated initial condition. $$ \frac{d y}{d x}+5 y=20, \quad y(0)=2 $$

It is sometimes possible to transform a nonexact differential equation \(M(x, y) d x+N(x, y) d y=0\) into an exact equation by multiplying it by an integrating factor \(\mu(x, y)\). In Problems \(37-42\) solve the given equation by verifying that the indicated function \(\mu(x, y)\) is an integrating factor. $$ \left(x^{2}+2 x y-y^{2}\right) d x+\left(y^{2}+2 x y-x^{2}\right) d y=0, \quad \mu(x, y)=(x+y)^{-2} $$

Solve the given differential equation subject to the indicated initial condition. $$ \left(1+x^{4}\right) d y+x\left(1+4 y^{2}\right) d x=0, \quad y(1)=0 $$

Solve the given differential equation subject to the indicated initial condition. $$ y d y=4 x\left(y^{2}+1\right)^{1 / 2} d x, \quad y(0)=1 $$

In Problems \(41-50\) solve the given differential equation subject to the indicated initial condition. $$ L \frac{d i}{d t}+R i=E: \quad L, R, \text { and } E \text { constants, } i(0)=i_{0} $$

In Problems \(41-50\) solve the given differential equation subject to the indicated initial condition. $$ y \frac{d x}{d y}-x=2 y^{2}, \quad y(1)=5 $$

Solve the given differential equation subject to the indicated initial condition. $$ \frac{d y}{d t}+t y=y, \quad y(1)=3 $$

Solve the given differential equation subject to the indicated initial condition. $$ \frac{d x}{d y}=4\left(x^{2}+1\right), \quad x\left(\frac{\pi}{4}\right)=1 $$