Can z1 = y1 + y2 and z2 = y1 - y2 Form a Fundamental Set of Solutions?

[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.

 

[lippyka]

a. If z1 and z2 are given by z1=y1+y2 and z2=y1-y2, then the second derivatives of z1 and z2 are: z1'' = y1'' + 2y2'' and z2'' = y1'' - 2y2''Substituting these into the ODE yields:z1''+(t)t'+q(t)z1= (y1''+(t)t'+q(t)y1) + 2(y2''+(t)t'+q(t)y2) = 0 + 0 = 0z2''+(t)t'+q(t)z2= (y1''+(t)t'+q(t)y1) - 2(y2''+(t)t'+q(t)y2) = 0 - 0 = 0Therefore, z1 and z2 are also solutions to the ODE, and since they are linear combinations of the original solutions, y1 and y2, they form a fundamental set of solutions. b. If y1 and y2 both achieve a maximum or minimum at the same point in I, then they must both have the same value at this point. This means that when y1 and y2 are combined, the resulting linear combination will also have the same value at the same point. However, for a fundamental set of solutions, the linear combinations must be linearly independent, which means that the two solutions must have different values at some point. Thus, if y1 and y2 achieve a maximum or minimum at the same point in I, then they cannot form a fundamental set of solutions on this interval.c. If y1 and y2 form a fundamental set of solutions on I, then they must be linearly independent, which means that their Wronskian must not be equal to 0. The Wronskian of y1 and y2 is given by:W(y1,y2)= y1*y2'-y1'*y2If y1 and y2 have a common inflection point, then their derivatives must be equal at this point, which would make the Wronskian equal to 0. Therefore, for y1 and y2 to