91Ó°ÊÓ

Solve the initial value problem. $$4 y^{\prime \prime}-4 y^{\prime}+y=0, \quad y(0)=-1, y^{\prime}(0)=2$$

Verify that the given function is a particular solution to the specified nonhomogeneous equation. Find the general solution and evaluate its arbitrary constants to find the unique solution satisfying the equation and the given initial conditions. $$\begin{array}{l} y^{\prime \prime}-2 y^{\prime}+y=x^{-1} e^{x}, x>0 \\ y_{\mathrm{p}}=x e^{x} \ln x, \quad y(1)=e, y^{\prime}(1)=0 \end{array}$$

Two linearly independent solutions \(y_{1}\) and \(y_{2}\) are given to the associated homogeneous equation of the variable-coefficient nonhomogeneous equation. Use the method of variation of parameters to find a particular solution to the nonhomogeneous equation. Assume \(x>0\) in each exercise. $$x^{2} y^{\prime \prime}+2 x y^{\prime}-2 y=x^{2}, \quad y_{1}=x^{-2}, y_{2}=x$$

Solve the initial value problem. $$3 y^{\prime \prime}+y^{\prime}-14 y=0, \quad y(0)=2, y^{\prime}(0)=-1$$

Two linearly independent solutions \(y_{1}\) and \(y_{2}\) are given to the associated homogeneous equation of the variable-coefficient nonhomogeneous equation. Use the method of variation of parameters to find a particular solution to the nonhomogeneous equation. Assume \(x>0\) in each exercise. $$x^{2} y^{\prime \prime}+x y^{\prime}-y=x, \quad y_{1}=x^{-1}, y_{2}=x$$

Solve the initial value problem. $$4 y^{\prime \prime}+4 y^{\prime}+5 y=0, \quad y(\pi)=1, y^{\prime}(\pi)=0$$

a. Show that there is no solution to the boundary value problem $$y^{\prime \prime}+4 y=0, \quad y(0)=0, y(\pi)=1$$ b. Show that there are infinitely many solutions to the boundary value problem $$y^{\prime \prime}+4 y=0, \quad y(0)=0, y(\pi)=0$$