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Question Number 128156 by DomaPeti last updated on 04/Jan/21

$$f{\left(x\right)}^{″}=\frac{1}{f{\left(x\right)}^2}$$$$f\left(x\right)=?$$

Answered by mr W last updated on 05/Jan/21

$$let{y}^{\prime }=u$$$$y^{″}=u\frac{du}{dy}=\frac{1}{y^2}$$$$udu=\frac{dy}{y^2}$$$$\frac{u^2}{2}=-\frac{1}{y}+C_1$$$$u=\frac{dy}{dx}=\pm \sqrt{2(C_1-\frac{1}{y})}$$$$\frac{dy}{\sqrt{C_1-\frac{1}{y}}}=\pm \sqrt{2}dx$$$$\int \frac{dy}{\sqrt{C_1-\frac{1}{y}}}=C_2\pm \sqrt{2}x$$$$......$$

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