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Question Number 144930 by alcohol last updated on 30/Jun/21

$$solve$$$$\frac{d^{2}x}{{dt}^2}=cosx$$

Answered by ArielVyny last updated on 30/Jun/21

$$-cosx$$

Answered by mr W last updated on 01/Jul/21

$$u=\frac{dx}{dt}$$$$\frac{d^{2}x}{{dt}^2}=\frac{du}{dt}=u\frac{du}{dx}=\cos x$$$$\frac{u^2}{2}=\sin x+C$$$$u=\frac{dx}{dt}=\pm \sqrt{2(\sin x+C)}$$$$\frac{dx}{\sqrt{\sin x+C}}=\pm \sqrt{2}dt$$$$\int \frac{dx}{\sqrt{\sin x+C}}=\pm \sqrt{2}\int dt$$$$\frac{dx}{\sqrt{\sin x+C}}=\pm \sqrt{2}t+C_1$$

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