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Question Number 181799 by amin96 last updated on 30/Nov/22

$$\mathit{s}\mathit{o}\mathit{l}\mathit{v}\mathit{e}\mathit{t}\mathit{h}\mathit{e}\mathit{d}\mathit{i}\mathit{f}\mathit{g}\mathit{e}\mathit{r}\mathit{e}\mathit{n}\mathit{t}\mathit{i}\mathit{a}\mathit{l}\mathit{e}\mathit{q}\mathit{u}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}$$$$\mathit{y}\sqrt{1+{\left({\mathit{y}}^{\prime }\right)}^2}={\mathit{y}}^{\prime }$$

Answered by mr W last updated on 30/Nov/22

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

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