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Question Number 184085 by Shrinava last updated on 02/Jan/23

$$({\mathrm{y}}^2+{\mathrm{xy}}^2){\mathrm{y}}^{{}^{\prime }}+{\mathrm{x}}^2-{\mathrm{yx}}^2=0$$$$$$

Answered by mr W last updated on 02/Jan/23

$$\frac{y^{2}dy}{y-1}=\frac{x^{2}dx}{x+1}$$$$\int \frac{y^{2}dy}{y-1}=\int \frac{x^{2}dx}{x+1}$$$$\int \frac{(y^2-1+1)dy}{y-1}=\int \frac{(x^2-1+1)dx}{x+1}$$$$\int (y+1+\frac{1}{y-1})dy=\int (x-1+\frac{1}{x+1})dx$$$$\Rightarrow \frac{y^2}{2}+y+\ln (y-1)=\frac{x^2}{2}-x+\ln (x+1)+C$$

Commented by Shrinava last updated on 02/Jan/23

$$\mathrm{thank}\mathrm{you}\mathrm{so}\mathrm{much}\mathrm{professor},\mathrm{cool}$$

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