How can i solve second order nonhomgeneous differential equation?

[m.mohamed]

I would like to solve a non-homogeneous differential equation with complex coefficient. The equation is in the form:



where Jx is not a function of z and k, w, u are constants

 

[_Eduardo_]

\[ E_x = j \omega \mu {J_x} + A(1-cos( \sqrt{k} t) + B \sin( \sqrt{k} t) \]

 

[m.mohamed]

Thanks for your replay Eduardo, can you send to me the method of solving?

 

[_Eduardo_]

My apologies for the k and z's forgotten in the previous message.


If Jx does not depend of z it can be treated as a constant in the differential equation, then the solution is the sum of one particular solution

\[E_{xp} = \frac{j \omega \mu\,J_x }{\sqrt{k}}\]

plus the solution of the homogeneous

\[E_{xh} = A\cos(\sqrt{k}z)+B\cos(\sqrt{k}z)\]


\[E_x = E_{xh} + E_{xp}\]

 

[CataM]

Eduardo, on your E_xp there is sqr("k"). In my approach there is none.

- - - Updated - - -



a=j·w·mu·Jx

 

[m.mohamed]

you mean in this case that the imaginary part of the particular solution dosn't matter, does it?
i think when the right hand side of the particular solution is complex, the method of solving will be different, is it right??

CataM, Ex is differentiated with z , so why you write the solution as a function of x?

 

[_Eduardo_]

Eduardo, on your E_xp there is sqr("k"). In my approach there is none.

- - - Updated - - -



a=j·w·mu·Jx

OMG! Yes, you are right.


But z instead x ;-)