Chapter 5, Problem 89CE

Consider the differential equation $d^{2}f\left(x\right)/d{x}^2=bf\left(x\right)$ . (a) Suppose that $f_1\left(x\right)$ and $f_2\left(x\right)$ are solutions. That is,
$\frac{d^2{f}_1\left(x\right)}{d{x}^2}=b{f}_1\left(x\right)$ and $\frac{d^2{f}_2\left(x\right)}{d{x}^2}=b{f}_2\left(x\right)$

Show that the equation also holds when the linear combination $A_1{f}_1\left(x\right)+A_2{f}_2\left(x\right)$ is inserted. (b) Suppose that $f_3\left(x\right)$ and $f_4\left(x\right)$ are solutions of $d^{2}f\left(x\right)/d{x}^2=b{f}^2\left(x\right)$ . Is $A_3{f}_3\left(x\right)+A_4{f}_4\left(x\right)$ a solution? Justify your answer.