How can I solve this first order linear differential equation? y &#x2032; </msu

Alissa Hancock 2022-07-16 Answered
How can I solve this first order linear differential equation?
y = 1 2 x + y
I have tried turning it into an inexact differential equation, but I get an integration factor μ ( x , y ) and I don't know how to apply it.
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Answers (2)

Gornil2
Answered 2022-07-17 Author has 20 answers
See v = 2 x + y ,, then 2 d v d x = v 2 ( 1 + d y d x )
The given equation reduces to
2 d v v 2 ( 2 v ) = d x .
Integrate. Use partial fractions for the left hand side. Can you take it from here?
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aggierabz2006zw
Answered 2022-07-18 Author has 5 answers
You can use a substitution to settle this differential equation. Let s(x)=x+y(x). Differentiating this gives us d s d x ( x ) = 1 + d y d x ( x ) = 1 + 1 2 s ( x ) .
Algebraic manipulation gives
d s d x ( x ) = 2 2 s ( x ) = 2 ( 1 1 s ( x ) ) d s d x ( x ) 1 1 s ( x ) = 2.
Integrating both sides with respect to x gives us
d s d x ( x ) 1 1 s ( x ) d x = 2 d x ln ( s ( x ) 1 ) + s ( x ) = 2 x + C
for a constant C. Rearrange for s(x) and substitute back.
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