Proving the superposition principle

1. Jan 24, 2008

bleucat

1. The problem statement, all variables and given/known data

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.

3. The attempt at a solution
I substituted (u + w) in for y, but how does that help me?

Thanks in advance for the help!

2. Jan 24, 2008

Rainbow Child

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

3. Jan 24, 2008

Mathdope

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

4. Jan 24, 2008

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.

5. Jan 24, 2008

bleucat

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

6. Jan 25, 2008

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?