solution-xdy-3y-e-x-dx-with-integration-factor-

Question Number 118349 by syamil last updated on 17/Oct/20

s o l u t i o n x d y + (3 y - e^{x}) d x

w i t h i n t e g r a t i o n f a c t o r

Answered by Dwaipayan Shikari last updated on 17/Oct/20

x d y + (3 y - e^{x}) d x = 0

x \frac{d y}{d x} + 3 y - e^{x} = 0

\frac{d y}{d x} + \frac{3 y}{x} = e^{x}

I . F = e^{\int \frac{3}{x}} = x^{3}

y . x^{3} = \int x^{3} e^{x}

y . x^{3} = x^{3} e^{x} - \int 3 x^{2} e^{x}

y . x^{3} = x^{3} e^{x} - 3 x^{2} e^{x} + 6 \int {x e}^{x}

y . x^{3} = x^{3} e^{x} - 3 x^{2} e^{x} + 6 {x e}^{x} - 6 e^{x} + C

y = e^{x} (1 - \frac{3}{x} + \frac{6}{x^{2}} - \frac{6}{x^{3}}) + {C x}^{- 3}