# Fundamental Solution ODE

1. May 22, 2010

### roeb

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

Verify that y1(x) = 1 and y2(x) = x^.5 are solutions of the following y y'' + (y')^.5 = 0. Then show that c1 + c2 x^.5 is not in general a solution of this equation.

2. Relevant equations

3. The attempt at a solution

I was able to show that both y1 and y2 are solutions to the DE. I found the Wronskian to be 1/(2 sqrt(x)) which is not equal to zero, so I was under the impression that this would mean that the two solutions would form a fundamental set of solution. Does anyone see why c1 + c2 x^.5 isn't a solution?

2. May 22, 2010

### Dick

3. May 22, 2010

### roeb

Does it matter if it's not linear in general? Boyce/Diprima's theorems don't seem to make note of whether or not the ODE must be linear for a set of fundamental solutions to be valid.

4. May 22, 2010

### Dick

I'd have to see the theorem, but linearity is the property that tells you if y1 and y2 are solutions to an ODE then so is c1*y1+c2*y2. I think you need it.