91Ó°ÊÓ

Solve the differential equation by the method of integrating factors. $$y^{\prime}+y=\cos \left(e^{x}\right)$$

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

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

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

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

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.

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.

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

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