Finding Second Order Linear Equation with x & x*ln(x) Solutions

[brad sue]

Hi ,
I am stuck with the following problem:

Find a second order linear homogeneous equation having the pair as a fundamental set of solutions:
y₁(x)=x , y₂(x)=x*ln(x).

My problem here is that I don't have the exponential form for the proposed solutions.

Thank you for your help

B.

 

[Tide]

You don't need an exponential form. Can you find a (single) linear combination of the function and its first and second derivatives that add to zero - for both functions?

 

[brad sue]

You don't need an exponential form. Can you find a (single) linear combination of the function and its first and second derivatives that add to zero - for both functions?

Ok you mean :
y1=x ----> (y1)'=1 -------> (y1)"=0
it gives: x*y' -y=0

y2(x)=x*ln(x) ----> (y2)'=1+ln(x) -----> (y2)"=1/x
it gives y*y"-(y'-1)=0

The second equation seems to be the good one since it is a second degree?
Am I right?

 

[Tide]

I don't think there is any need to go nonlinear. Try this:

x y'' - x y' + y = 0