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Question Number 116053 by Study last updated on 30/Sep/20

$$y\frac{dy}{dx}=1+x^{2}y=?$$

Answered by Dwaipayan Shikari last updated on 30/Sep/20

$$\mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}}=1+{\mathrm{x}}^2$$$$\int \mathrm{ydy}=\int 1+{\mathrm{x}}^2\mathrm{dx}$$$$\frac{{\mathrm{y}}^2}{2}=\mathrm{x}+\frac{{\mathrm{x}}^3}{3}+\mathrm{C}$$$$\mathrm{y}=\pm \sqrt{2\mathrm{x}+\frac{{2\mathrm{x}}^3}{3}+2\mathrm{C}}$$

Answered by Bird last updated on 30/Sep/20

$${yy}^{{}^{\prime }}=1+x^2\Rightarrow \int {yy}^{{}^{\prime }}dx=x+\frac{x^3}{3}+c$$$$\Rightarrow \frac{1}{2}{y}^2=x+\frac{x^3}{3}+c\Rightarrow$$$$y^2=2x+\frac{2x^3}{3}+2c\Rightarrow$$$$y=\stackrel{-}{+}\sqrt{2x+\frac{2x^3}{3}+\lambda }(\lambda =2c)$$

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