91Ó°ÊÓ

Solve the given differential equation by separation of variables. $$ x \sqrt{1-y^{2}} d x=d y $$

In Problems 1-40 find the general solution of the given differential equation. State an interval on which the general solution is defined. $$ (x+2)^{2} \frac{d y}{d x}=5-8 y-4 x y $$

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. $$ -y^{2} d x+\left(x^{2}+x y\right) d y=0, \quad \mu(x, y)=\frac{1}{x^{2} y} $$

Solve the given differential equation by separation of variables. $$ y\left(4-x^{2}\right)^{12} d y=\left(4+y^{2}\right)^{12} d x $$

Solve the given differential equation by separation of variables. $$ \left(e^{x}+e^{-x}\right) \frac{d y}{d x}=y^{2} $$

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. $$ (-x y \sin x+2 y \cos x) d x+2 x \cos x d y=0, \quad \mu(x, y)=x y $$

In Problems 1-40 find the general solution of the given differential equation. State an interval on which the general solution is defined. $$ y^{\prime}=(10-y) \cosh x $$

In Problems 1-40 find the general solution of the given differential equation. State an interval on which the general solution is defined. $$ d x=\left(3 e^{y}-2 x\right) d y $$

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. $$ y(x+y+1) d x+(x+2 y) d y=0, \quad \mu(x, y)=e^{x} $$

Solve the given differential equation by separation of variables. $$ (x+\sqrt{x}) \frac{d y}{d x}=y+\sqrt{y} $$