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Question Number 181219 by Mastermind last updated on 23/Nov/22

$$\mathrm{Solve}\mathrm{the}\mathrm{D}.\mathrm{E}$$$$\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}-\mathrm{y}=\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}$$$$$$$$.$$

Answered by qaz last updated on 23/Nov/22

$$x\frac{dy}{dx}-y=x^2\frac{d}{dx}\left(\frac{y}{x}\right)=\sqrt{x^2+y^2}$$$$\Rightarrow x\frac{d}{dx}\left(\frac{y}{x}\right)=\sqrt{1+{\left(\frac{y}{x}\right)}^2}$$$$\Rightarrow \frac{d\left(\frac{y}{x}\right)}{\sqrt{1+{\left(\frac{y}{x}\right)}^2}}=\frac{dx}{x}$$$$\Rightarrow ln(\left(\frac{y}{x}\right)+\sqrt{1+{\left(\frac{y}{x}\right)}^2})=lnx+C$$$$\Rightarrow y+\sqrt{x^2+y^2}={Cx}^2$$

Commented by Mastermind last updated on 29/Nov/22

$$\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{understand}\mathrm{this}...\mathrm{kindly}\mathrm{show}$$$$\mathrm{the}\mathrm{steps}$$

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