What is the solution for (x+4)(y^2+1)dx+y(x^2+3x+2)dy=0 ?

The solution for (x+4)(y^2+1)dx+y(x^2+3x+2)dy=0 is

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(x+4)(y^2+1)dx+y(x^2+3x+2)dy=0

(x + 4) (y^{2} + 1) d x + y (x^{2} + 3 x + 2) d y = 0

y = \frac{e ^{2 c_{1}} x ^{4} + 8 e ^{2 c_{1}} x ^{3} + 24 e ^{2 c_{1}} x ^{2} + 32 e ^{2 c_{1}} x + 16 e ^{2 c_{1}} - x ^{6} - 6 x ^{5} - 15 x ^{4} - 20 x ^{3} - 15 x ^{2} - 6 x - 1}{x ^{6} + 6 x ^{5} + 15 x ^{4} + 20 x ^{3} + 15 x ^{2} + 6 x + 1}, y = - \frac{e ^{2 c_{1}} x ^{4} + 8 e ^{2 c_{1}} x ^{3} + 24 e ^{2 c_{1}} x ^{2} + 32 e ^{2 c_{1}} x + 16 e ^{2 c_{1}} - x ^{6} - 6 x ^{5} - 15 x ^{4} - 20 x ^{3} - 15 x ^{2} - 6 x - 1}{x ^{6} + 6 x ^{5} + 15 x ^{4} + 20 x ^{3} + 15 x ^{2} + 6 x + 1}

Solution steps

(x + 4) (y^{2} + 1) d x + y (x^{2} + 3 x + 2) d y = 0

Solve separable ODE:

y = \frac{e ^{2 c_{1}} x ^{4} + 8 e ^{2 c_{1}} x ^{3} + 24 e ^{2 c_{1}} x ^{2} + 32 e ^{2 c_{1}} x + 16 e ^{2 c_{1}} - x ^{6} - 6 x ^{5} - 15 x ^{4} - 20 x ^{3} - 15 x ^{2} - 6 x - 1}{x ^{6} + 6 x ^{5} + 15 x ^{4} + 20 x ^{3} + 15 x ^{2} + 6 x + 1}, y = - \frac{e ^{2 c_{1}} x ^{4} + 8 e ^{2 c_{1}} x ^{3} + 24 e ^{2 c_{1}} x ^{2} + 32 e ^{2 c_{1}} x + 16 e ^{2 c_{1}} - x ^{6} - 6 x ^{5} - 15 x ^{4} - 20 x ^{3} - 15 x ^{2} - 6 x - 1}{x ^{6} + 6 x ^{5} + 15 x ^{4} + 20 x ^{3} + 15 x ^{2} + 6 x + 1}

y = \frac{e ^{2 c_{1}} x ^{4} + 8 e ^{2 c_{1}} x ^{3} + 24 e ^{2 c_{1}} x ^{2} + 32 e ^{2 c_{1}} x + 16 e ^{2 c_{1}} - x ^{6} - 6 x ^{5} - 15 x ^{4} - 20 x ^{3} - 15 x ^{2} - 6 x - 1}{x ^{6} + 6 x ^{5} + 15 x ^{4} + 20 x ^{3} + 15 x ^{2} + 6 x + 1}, y = - \frac{e ^{2 c_{1}} x ^{4} + 8 e ^{2 c_{1}} x ^{3} + 24 e ^{2 c_{1}} x ^{2} + 32 e ^{2 c_{1}} x + 16 e ^{2 c_{1}} - x ^{6} - 6 x ^{5} - 15 x ^{4} - 20 x ^{3} - 15 x ^{2} - 6 x - 1}{x ^{6} + 6 x ^{5} + 15 x ^{4} + 20 x ^{3} + 15 x ^{2} + 6 x + 1}

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What is the solution for (x+4)(y^2+1)dx+y(x^2+3x+2)dy=0 ?
The solution for (x+4)(y^2+1)dx+y(x^2+3x+2)dy=0 is