91Ó°ÊÓ

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x. $$ \frac{d y}{d x}-\frac{y^{2}-y}{\sin x}=0 $$

In each part, verify that the functions are solutions of the differential equation by substituting the functions into the equation. $$ \begin{array}{l}{y^{\prime \prime}+y^{\prime}-2 y=0} \\ {\text { (a) } e^{-2 x} \text { and } e^{x}} \\ {\text { (b) } c_{1} e^{-2 x}+c_{2} e^{x}\left(c_{1}, c_{2} \text { constants }\right)}\end{array} $$

Solve the initial-value problem. $$ \frac{d y}{d t}+y=2, \quad y(0)=1 $$

Use Euler's Method with the given step size \(\Delta x\) or \(\Delta t\) to approximate the solution of the initial-value problem over the stated interval. Present your answer as a table and as a graph. $$d y / d t=e^{-y}, y(0)=0,0 \leq t \leq 1, \Delta t=0.1$$

In each part, verify that the functions are solutions of the differential equation by substituting the functions into the equation. $$ \begin{array}{l}{y^{\prime \prime}-y^{\prime}-6 y=0} \\ {\text { (a) } e^{-2 x} \text { and } e^{3 x}} \\ {\text { (b) } c_{1} e^{-2 x}+c_{2} e^{3 x}\left(c_{1}, c_{2} \text { constants }\right)}\end{array} $$

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x. $$ y-\frac{d y}{d x} \sec x=0 $$

In each part, verify that the functions are solutions of the differential equation by substituting the functions into the equation. $$ \begin{array}{l}{y^{\prime \prime}-4 y^{\prime}+4 y=0} \\ {\text { (a) } e^{2 x} \text { and } x e^{2 x}} \\ {\text { (b) } c_{1} e^{2 x}+c_{2} x e^{2 x}\left(c_{1}, c_{2} \text { constants }\right)}\end{array} $$

Consider the initial-value problem $$ y^{\prime}=\sin \pi t, \quad y(0)=0 $$ Use Euler's Method with five steps to approximate \(y(1)\)

Solve the initial-value problem by separation of variables. $$ y^{\prime}=\frac{3 x^{2}}{2 y+\cos y}, \quad y(0)=\pi $$

Determine whether the statement is true or false. Explain your answer. If the first-order linear differential equation $$ \frac{d y}{d x}+p(x) y=q(x) $$ has a solution that is a constant function, then \(q(x)\) is a constant multiple of \(p(x) .\)