General solution of the differential equation y'=y/x+xe^x?

rustenig

rustenig

Answered question

2022-09-06

General solution of the differential equation y = y x + x e x ?

Answer & Explanation

Raphael Singleton

Raphael Singleton

Beginner2022-09-07Added 19 answers

We seek a solution to the First Order ODE:

y = y x + x e x

We can use an integrating factor when we have a First Order Linear non-homogeneous Ordinary Differential Equation of the form;

d y d x + P ( x ) y = Q ( x )

So rewrite the equations in standard form as:

d y d x - y x = x e x ... . . [ 1 ]

Then the integrating factor is given by;

I = e P ( x ) d x
    = exp (   - 1 x   d x )
    = exp ( - ln x )
    = 1 x

And if we multiply the DE [1] by this Integrating Factor, I, we will have a perfect product differential;

1 x d y d x - y x 2 = e x

d d x ( y x ) = e x

Which we can directly integrate to get:

y x =   e x + C

y x = e x + C

y = x e x + C x

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