91Ó°ÊÓ

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

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

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. \(y^{\prime \prime}-y^{\prime}-6 y=0\) (a) \(e^{-2 x}\) and \(e^{3 x}\) (b) \(c_{1} e^{-2 x}+c_{2} e^{3 x}\left(c_{1}, c_{2}\right.\) constants)

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

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

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

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. $$ y^{\prime}-x e^{y}=2 e^{y}, \quad y(0)=0 $$

In each part, verify that the functions are solutions of the differential equation by substituting the functions into the equation. \(y^{\prime \prime}-8 y^{\prime}+16 y=0\) (a) \(e^{4 x}\) and \(x e^{4 x}\) (b) \(c_{1} e^{4 x}+c_{2} x e^{4 x}\left(c_{1}, c_{2}\right.\) constants \()\)