I'm wondering if there's a simple proof that solutions to non-linear differential equations do not satisfy the superposition principle?
Some explicit examples would also be great.
Cheers!
I'm wondering if there's a simple proof that solutions to non-linear differential equations do not satisfy the superposition principle?
Some explicit examples would also be great.
Cheers!
For such cases the simplest example that comes to mind usually works. Consider d.e. $y'=y^2$. It has solutions $y_c(x)=\frac 1{c-x}\,$ for $c\in \mathbb R$. But functions $y_1+y_2$ and $2y_1$ are not solutions.
But it cannot be said that all solutions of any nonlinear equation do not satisfy the superposition principle. Consider, for example, equation $(y'')^2=0$.
Superposition theorem only works for linear differential equations. Reason is For a non-linear differential equation we are using transformations that would be the solution. Again if we think for other solution and their linear combination seems not at all a solution.