# Fundamental Set of solutions

1. Mar 13, 2008

### chobo86

Suppose that p and q are continuous on some open interval I and suppose that y1 and y2 are solutions o the ode
y''+(t)t'+q(t)y=0

a. Suppose that y1 , y2 is a fundamental set of solutions. Prove that z1, z2 given by z1=y1+y2, z2=y1-y2 is also a fundamental set of solutions.

b. prove that if y1 and y2 achieve a maximu or a minimumat the ame point in I, then they cannot form a fundamental set of solutions on this interval

c. Prove that if y1 and y2 form a fundamental set of solutions on I, then they cannot have a common inflection point in I, unless p and q are both 0 at this point

d. if 0$$\in$$I show that y(t)=t^3 cannot be a solution of the ODE on I.
1. The problem statement, all variables and given/known data

2. Mar 11, 2009

### hsong9

c. suppose y1 and y2 have a common inflection point in I...
so.. y''= 0 and p(t)y' + q(t)y = 0 and consider W = |y1'y2 - y1'y2|

If fundamental set of solutions... W != 0.

3. Mar 11, 2009

### John Creighto

This is trivial. Substitute z into the ode and see what you get. See what algebra you can do to show something you know is true. Haven't tried it but it should work.

4. Mar 11, 2009

### John Creighto

Should this:
y''+(t)t'+q(t)y=0

be

y''+(t)y'+q(t)y=0