Chapter 3, Problem 1RQ

To determine

To define: The superposition or linearity principle and state whether it holds for the nth order ODEs.

Explanation of Solution

The superposition or linearity principle states that the addition of the given solutions or the product by a constant with the solutions results in further solutions.

In other words, any linear combination of solutions is also a solution.

That is, if $y_1\left(x\right)\text{ and }{y}_2\left(x\right)$ are the solutions of the ODE $a{y}^n+b{y}^{n-1}+\cdots +c=0$, then $m\left(y_1\left(x\right)+y_2\left(x\right)\right),m\text{ is any scalar}$ is also a solution for the given ODE.

The superposition only holds for the nth order homogeneous ODEs.