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Question Number 91812 by niroj last updated on 03/May/20

$$\mathrm{Solve}{\mathrm{clairaut}}^{\prime }\mathrm{s}\mathrm{equation}\mathrm{and}\mathrm{find}$$$$\mathrm{general}\mathrm{and}\mathrm{singular}\mathrm{solution}:$$$$\left(\mathrm{i}\right)\mathrm{y}=\mathrm{px}+{\mathrm{p}}^{\mathrm{n}}$$$$\left(\mathrm{ii}\right)(\mathrm{y}+1)\mathrm{p}-{\mathrm{xp}}^2+2=0$$$$$$$$$$

Commented by john santu last updated on 03/May/20

$$\left(1\right)putxy=Y;x=X$$$$x\frac{dy}{dx}+y=\frac{dY}{dx}=P;dx=dX$$$$y+px=P,wherep=\frac{dy}{dx}$$$$\Rightarrow y=px+p^n$$$$y=P-y-p^n\Rightarrow 2y=P-p^n$$$$\frac{Y}{x}=P-{\left(\frac{P-y}{x}\right)}^n$$$$Y=Px-\frac{{(P-\frac{Y}{x})}^n}{x^{n-1}}$$$$continue...$$

Commented by john santu last updated on 03/May/20

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