What is a general solution to the differential equation y'+2xy=2x^3?

frobirrimupyx 2022-09-14 Answered
What is a general solution to the differential equation y + 2 x y = 2 x 3 ?
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Answers (1)

Anabelle Guzman
Answered 2022-09-15 Author has 14 answers
y'+2xy=2x3
this is not separable. we can use an integrating factor η ( x )
here η = exp ( 2 x   d x ) = e x 2
so
e x 2 y + e x 2 2 x y = e x 2 2 x 3
or
d d x ( e x 2 y ) = e x 2 2 x 3
e x 2 y =   e x 2 2 x 3 d x
e x 2 y =   d d x ( e x 2 ) x 2 d x
using IBP
e x 2 y = x 2 e x 2 -   e x 2 d d x ( x 2 ) d x
e x 2 y = x 2 e x 2 - 2   x e x 2 d x
and spotting the pattern in the integrand
e x 2 y = x 2 e x 2 - 2   d d x ( 1 2 e x 2 ) d x
e x 2 y = x 2 e x 2 - e x 2 + C
y = x 2 - 1 + C e - x 2

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