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# How can i solve second order nonhomgeneous differential equation?

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#### m.mohamed

##### Member level 1
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_

##### Full Member level 5
$E_x = j \omega \mu {J_x} + A(1-cos( \sqrt{k} t) + B \sin( \sqrt{k} t)$

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#### m.mohamed

##### Member level 1
Thanks for your replay Eduardo, can you send to me the method of solving?

#### _Eduardo_

##### Full Member level 5
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 Exp there is sqr("k"). In my approach there is none.

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a=j·w·mu·Jx

#### m.mohamed

##### Member level 1

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_

##### Full Member level 5
Eduardo, on your Exp there is sqr("k"). In my approach there is none.

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a=j·w·mu·Jx

OMG! Yes, you are right.