Fundamental Solutions for ODEs with Continuous Coefficients

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

Homework Statement

 

[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.

 

[John Creighto]

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.

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.

 

[John Creighto]

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

be

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