xy'−3y=x−1? Solve

Mutignaniz2 2022-09-12 Answered
x y - 3 y = x - 1 Solve
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Answers (1)

Griffin Dean
Answered 2022-09-13 Author has 13 answers
Divide all terms by x
d y d x - 3 y x = x - 1 x
The ODE is now in a form where it can be solved using an integrating factor
Let μ = exp - 3 x = exp ( - 3 ln x ) = exp ln ( 1 x 3 ) = 1 x 3
Now multiply the whole equation by μ
1 x 3 d y d x - 3 y x 4 = x - 1 x 4
The LHS factors as a derivative and we can integrate the RHS
d d x ( y x 3 ) d x = x - 3 - x - 4 d x
y x 3 = - 1 2 x - 2 + 1 3 x - 3 + a
Solve for y
y = y ( x ) = a x 3 - 1 2 x + 1 3

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