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The general solution of differential equation $e^{\frac{1}{2}\left(\frac{d y}{d x}\right)}=3^x$ is (where $C$ is a constant of integration.)
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$y=x^2 \log 3+C$
$\begin{aligned} & e^{\frac{1}{2}\left(\frac{d y}{d x}\right)}=3^x \Rightarrow \frac{1}{2} \frac{d y}{d x}=\log _e 3^x=x \log _e 3 \\ & \Rightarrow \frac{d y}{d x}=\left(2 \log _e 3\right) x \\ & \Rightarrow y=2 \log _e 3 \times \frac{x^2}{2}+c \\ & \Rightarrow y=x^2 \log _e 3+c\end{aligned}$
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