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Question Number 108406 by pticantor last updated on 16/Aug/20

$$$$$$$$$$\mathit{s}\mathit{o}\mathit{l}\mathit{v}\mathit{e}\mathit{t}\mathit{h}\mathit{e}\mathit{d}\mathit{i}\mathit{f}\mathit{f}\mathit{e}\mathit{r}\mathit{e}\mathit{n}\mathit{t}\mathit{i}\mathit{a}\mathit{l}$$$$\mathit{e}\mathit{q}\mathit{u}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}$$$${\mathit{y}}^{{}^{\prime }}-2e^x\mathit{y}=2e^x\sqrt{\mathit{y}}$$

Answered by Dwaipayan Shikari last updated on 16/Aug/20

$$\frac{dy}{dx}=2e^x(y+\sqrt{y})$$$$\int \frac{dy}{y+\sqrt{y}}=\int 2e^{x}dx$$$$\int \frac{dy}{\sqrt{y}(1+\sqrt{y})}dy=2e^x+C$$$$\int \frac{2t}{t(1+t)}dt=2e^x+Cy=t^2,1=2t\frac{dt}{dy}$$$$2log(1+t)=2e^x+C$$$$log(1+t)=e^x+\frac{C}{2}$$$$log(1+\sqrt{y})=e^x+\frac{C}{2}$$

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