Thread: 2nd order ODE

Hi ,
I am stuck with the following problem (to turn tomorrow):

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 dont have the exponential form for the proposed solutions.

Thank you for your help

B.

Originally Posted by braddy

Hi ,
I am stuck with the following problem (to turn tomorrow):

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 dont have the exponential form for the proposed solutions.

Thank you for your help

B.

A second order linear homogeneous ODE has the form:



Now



is a solution so differentiating twice and substituting the derivatives of
into the ODE gives:



or:



Similarly



is also a solution so:



Now substituting from into this last
equation gives:



So rearranging we find:



and:



RonL

Originally Posted by CaptainBlack

A second order linear homogeneous ODE has the form:



Now



is a solution so differentiating twice and substituting the derivatives of
into the ODE gives:



or:



Similarly



is also a solution so:



Now substituting from into this last
equation gives:



So rearranging we find:



and:



RonL

Thank you very much Captain,
Because we use a different form in class, the exponential form for the solution (y=e(rx)), I was a little bit confused by this problem . Now I get it!

B