Chapter 4.1, Problem 13E

In Problems 13 and 14 the given two-parameter family is a solution of the indicated differential equation on the interval (−∞, ∞). Determine whether a member of the family can be found that satisfies the boundary conditions.

13. $y=c_1{e}^x\cos x+c_2{e}^x\sin x;y^{″}-2y^{\prime }+2y=0$

$\begin{array}{l}\begin{array}{ccc}(a)& y(0)=1,& y^{\prime }(\pi )\end{array}=0\\ \begin{array}{ccc}(b)& y(0)=1,& y^{\prime }(\pi )=-1\end{array}\\ \begin{array}{ccc}(c)& y(0)=1,& y^{\prime }(\pi /2)=1\end{array}\\ \begin{array}{ccc}(d)& y(0)=0,& y^{\prime }(\pi )=0.\end{array}\end{array}$