The general solution of differential equation $e^{\frac{1}{2}\left(\frac{d y}{d x}\right)}=3^x$ is (where…

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.)
  1. $y=x \log 3+C$
  2. $y=x^2 \log 3+C$
  3. $y=2 x \log 3+C$
  4. $x=(\log 3) y^2+C$

Solution

$\begin{aligned} & e^{\frac{1}{2}\left(\frac{d y}{d x}\right)}=3^x \Rightarrow \frac{1}{2} \frac{d y}{d x}=\log _e 3^x=x \log _e 3 \\ & \Rightarrow \frac{d y}{d x}=\left(2 \log _e 3\right) x \\ & \Rightarrow y=2 \log _e 3 \times \frac{x^2}{2}+c \\ & \Rightarrow y=x^2 \log _e 3+c\end{aligned}$

Asked in: MHT CET 2022 (07 Aug Shift 1)

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