The solution of the differential equation $\mathrm{e}^{\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)}=x+1 ; y(0)=5$, $x \in(-1, \infty)$

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Aicharya · Accepted Answer

$\begin{aligned} & \mathrm{e}^{\frac{\mathrm{d} y}{\mathrm{~d} x}}=x+1 \ & \Rightarrow \frac{\mathrm{d} y}{\mathrm{~d} x}=\log _{\mathrm{e}}(x+1) \ & \Rightarrow \int \mathrm{d} y=\int \log _{\mathrm{e}}(x+1) \mathrm{d} x \ & \Rightarrow y=(x+1) \log _{\mathrm{e}}(x+1)-(x+1)+C 	ext { [integrating by parts] } \ & 	ext { putting } x=0 	ext { and } y=5 	ext { we get } C=6 \ & \Rightarrow y=(x+1) \log _{\mathrm{e}}(x+1)-x+5\end{aligned}$

The solution of the differential equation $\mathrm{e}^{\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)}=x+1 …

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