Proving the superposition principle

[bleucat]

Homework Statement

Hi everyone.
I am trying to prove the superposition principle for linear homogeneous equations, which states that if u(t) and w(t) are solutions to y' + p(t)y = 0, then u(t) + w(t) and k(u(t)) are also solutions for any constant k.

The Attempt at a Solution

I substituted (u + w) in for y, but how does that help me?

Thanks in advance for the help!

 

[Rainbow Child]

Use the fact that $u(t),w(t)$ satisfy the ODE.

 

[Mathdope]

As in, what differentiation property makes u + w a solution if u, w are?

 

[Rainbow Child]

The statement $u(t),w(t)$: solutions of the ODE means

$$u'(t)+p(t)\,u(t)=0, \, w'(t)+p(t)\,w(t)=0$$

Put $y'(t)=u'(t)+w'(t)$ in the original ODE and use the above equations.

 

[bleucat]

Hah, I got it. That wasn't bad at all.
Thanks!

 

[HallsofIvy]

Now, can you do it for ku(t)? And, can you prove that those two together show that
au(t)+ bv(t) is a solution for any numbers a, b, as long as u(t) and v(t) are solutions?