Chapter 7.1, Problem 52E

(Principle of superposition) Prove that if $y_1\left(x\right)$ and $y_2\left(x\right)$ are solutions to a linear homogeneous differential equation, $y^{″}+p\left(x\right)y^{\prime }+q\left(x\right)y=0$ , then the function $y\left(x\right)=c_1{y}_1\left(x\right)+c_2{y}_2\left(x\right)$ , where $c_1$ and $c_2$ are constants, is also a solution.