I am aware that the solution to a homogeneous first order differential equation of the form

Kristen Stokes 2022-07-07 Answered
I am aware that the solution to a homogeneous first order differential equation of the form d y d x = p ( x ) y can be obtained by simply by rearranging to:
d y y = p ( x ) d x
Then it is simply a question of integrating both sides and the answer is straightforward. Now what would happen if RHS had a constant, how a can find a particular solution to this case: d y d x = p ( x ) y + C
I know that the general solution would be the sum of the homogeneous equation and the particular solution
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Answers (1)

Kiana Cantu
Answered 2022-07-08 Author has 22 answers
Simply rearrange :
d y d x y p ( x ) = C
Now multiply both the sides by e p ( x ) d x
d y d x e p ( x ) d x y p ( x ) e p ( x ) d x = C e p ( x ) d x
This LHS is nothing else :
d y d x e p ( x ) d x y p ( x ) e p ( x ) d x = d d x ( y e p ( x ) d x )
Now our differential equation becomes :
d d x ( y e p ( x ) d x ) = C e p ( x ) d x
Hence:
y = C e p ( x ) d x d x e p ( x ) d x
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