91Ó°ÊÓ

State the order of the differential equation, and confirm that the functions in the given family are solutions. $$ \begin{array}{l}{\text { (a) }(1+x) \frac{d y}{d x}=y ; y=c(1+x)} \\ {\text { (b) } y^{\prime \prime}+y=0 ; y=c_{1} \sin t+c_{2} \cos t}\end{array} $$

State the order of the differential equation, and confirm that the functions in the given family are solutions. $$ \begin{array}{l}{\text { (a) } 2 \frac{d y}{d x}+y=x-1 ; \quad y=c e^{-x / 2}+x-3} \\ {\text { (b) } y^{\prime \prime}-y=0 ; \quad y=c_{1} e^{t}+c_{2} e^{-t}}\end{array} $$

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x. $$ \left(1+x^{4}\right) \frac{d y}{d x}=\frac{x^{3}}{y} $$

Solve the differential equation by the method of integrating factors. $$ 2 \frac{d y}{d x}+4 y=1 $$

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x. $$ \left(2+2 y^{2}\right) y^{\prime}=e^{x} y $$

Solve the differential equation by the method of integrating factors. $$ \left(x^{2}+1\right) \frac{d y}{d x}+x y=0 $$

True-False Determine whether the statement is true or false. Explain your answer. The equation $$\left(\frac{d y}{d x}\right)^{2}=\frac{d y}{d x}+2 y$$ is an example of a second-order differential equation.

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x. $$ y^{\prime}=-x y $$

Solve the differential equation by the method of integrating factors. $$ \frac{d y}{d x}+y+\frac{1}{1-e^{x}}=0 $$

True-False Determine whether the statement is true or false. Explain your answer. The differential equation $$\frac{d y}{d x}=2 y+1$$ has a solution that is constant.