What is the solution for (5+x)y^'=8y ?

The solution for (5+x)y^'=8y is

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(5+x)y^'=8y

(5 + x) y^{^{'}} = 8 y

y = 390625 e^{c_{1}} + 625000 e^{c_{1}} x + 437500 e^{c_{1}} x^{2} + 175000 e^{c_{1}} x^{3} + 43750 e^{c_{1}} x^{4} + 7000 e^{c_{1}} x^{5} + 700 e^{c_{1}} x^{6} + 40 e^{c_{1}} x^{7} + e^{c_{1}} x^{8}

Solution steps

(5 + x) y^{'} (x) = 8 y

Solve separable ODE:

y = 390625 e^{c_{1}} + 625000 e^{c_{1}} x + 437500 e^{c_{1}} x^{2} + 175000 e^{c_{1}} x^{3} + 43750 e^{c_{1}} x^{4} + 7000 e^{c_{1}} x^{5} + 700 e^{c_{1}} x^{6} + 40 e^{c_{1}} x^{7} + e^{c_{1}} x^{8}

y = 390625 e^{c_{1}} + 625000 e^{c_{1}} x + 437500 e^{c_{1}} x^{2} + 175000 e^{c_{1}} x^{3} + 43750 e^{c_{1}} x^{4} + 7000 e^{c_{1}} x^{5} + 700 e^{c_{1}} x^{6} + 40 e^{c_{1}} x^{7} + e^{c_{1}} x^{8}

(3 x^{2} + 5 x) \frac{d y}{d x} + 5 y = - 6 x y

\frac{d y}{d x} = \frac{- 1}{x ^{2} y ^{2}}

y^{^{'}} = \frac{8}{x} - \frac{6}{x} y

\frac{d y}{d x} = (\frac{x ^{2}}{y})

- y^{3} + y^{^{'}} + 0.5 y = 0