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Question Number 64973 by Tawa1 last updated on 23/Jul/19

Answered by mr W last updated on 23/Jul/19

$$\left(1\right)$$$$\frac{dy}{dx}=u$$$$\frac{d^{2}y}{{dx}^2}=\frac{du}{dx}=\frac{du}{dy}\times \frac{dy}{dx}=u\frac{du}{dy}$$$$u\frac{du}{dy}=\frac{4}{y^3}$$$$udu=\frac{4dy}{y^3}$$$$\int udu=\int \frac{4dy}{y^3}$$$$\frac{u^2}{2}=-\frac{2}{y^2}+C_1$$$$u^2=\frac{c_1{y}^2-4}{y^2}$$$$u=\frac{dy}{dx}=\frac{\sqrt{c_1{y}^2-4}}{y}$$$$\frac{ydy}{\sqrt{c_1{y}^2-4}}=dx$$$$\int \frac{ydy}{\sqrt{c_1{y}^2-4}}=\int dx$$$$\int \frac{d\left(c_1{y}^2\right)}{\sqrt{c_1{y}^2-4}}=2c_{1}x+C_2$$$$\Rightarrow 2\sqrt{c_1{y}^2-4}=2c_{1}x+C_2$$$$\Rightarrow \sqrt{c_1{y}^2-4}=c_{1}x+c_2$$

Commented by Tawa1 last updated on 23/Jul/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

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