Question Number 77473 by BK last updated on 06/Jan/20
Commented by mr W last updated on 06/Jan/20
$${y}=\frac{{dy}}{{dx}}+\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }+\frac{{d}^{\mathrm{3}} {y}}{{dy}^{\mathrm{3}} }+... \\ $$$$\frac{{dy}}{{dx}}=\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }+\frac{{d}^{\mathrm{3}} {y}}{{dy}^{\mathrm{3}} }+... \\ $$$$\mathrm{2}\frac{{dy}}{{dx}}=\frac{{dy}}{{dx}}+\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }+\frac{{d}^{\mathrm{3}} {y}}{{dy}^{\mathrm{3}} }+... \\ $$$$\mathrm{2}\frac{{dy}}{{dx}}={y} \\ $$$$\frac{\mathrm{2}{dy}}{{y}}={dx} \\ $$$$\mathrm{2}\int\frac{{dy}}{{y}}=\int{dx} \\ $$$$\mathrm{2ln}\:{y}={x}+{C}_{\mathrm{1}} \\ $$$$\Rightarrow{y}={Ce}^{\frac{{x}}{\mathrm{2}}} \\ $$
Commented by BK last updated on 06/Jan/20
$$\mathrm{thanks} \\ $$