91Ó°ÊÓ

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

Determine whether the statement is true or false. Explain your answer. We expect the general solution of the differential equation $$ \frac{d^{3} y}{d x^{3}}+3 \frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}+4 y=0 $$ to involve three arbitrary constants.

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. $$d y / d x=\sqrt[3]{y}, y(0)=1,0 \leq x \leq 4, \Delta x=0.5$$

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

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of \(x\). $$y^{\prime}-(1+x)\left(1+y^{2}\right)=0$$

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

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. $$d y / d x=x-y^{2}, y(0)=1,0 \leq x \leq 2, \Delta x=0.25$$

Determine whether the statement is true or false. Explain your answer. If every solution to a differential equation can be expressed in the form \(y=A e^{x+b}\) for some choice of constants \(A\) and \(b,\) then the differential equation must be of second order.

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

Solve the initial-value problem. $$\frac{d y}{d x}-2 x y=2 x, \quad y(0)=3$$