# 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$$\Iin$$ show that y(t)=t^3 cannot be a solution of the ODE on I.