Answer to Question #296832 in Classical Mechanics for Shah

Question

Show that superposition principle does not hold for a non linear DE like; d²x/d²t + cx² = 0 where c is constant.

Answer 1

Let us assume $x_1$ and $x_2$ both are solution of DE

$$d²x/d²t + cx² = 0$$

So

$$d²x_1/d²t + cx_1^2= 0,\quad d²x_2/d²t + cx_2^2= 0$$

Now, we consider a superposition of these solutions

$$x=x_1+x_2$$

We get

$$d²(x_1+x_2)/d²t + c(x_1+x_2)^2$$

$$=d²x_1/d²t + cx_1^2+d²x_2/d²t + cx_2^2+2cx_1x_2\\ =2cx_1x_2\neq0$$

Thus, the superposition principle does not hold for a non linear DE.