Question

The solution of $\frac{dy}{dx}=3^{x+y}$ is

• A. $3^{x+y}=c$
• B. $3^x+3^y=c$
• C. $3^{x-y}=c$
• D. $3^x+3^{-y}=c$

Answer 1

The correct option is D

$3^x+3^{-y}=c$

Explanation for the correct option:

Find the solution of given differential equation

Given: $\frac{dy}{dx}=3^{x+y}$

$$\begin{gathered} \Rightarrow \frac{dy}{dx}=3^x.3^y \\ \Rightarrow 3^{-y}dy=3^{x}dx \end{gathered}$$

On integrating both sides, we get
$$\begin{gathered} \int 3^{-y}dy=\int 3^{x}dx\left[\because \int a^x\mathrm{dx}=\frac{a^x}{\ln \left(a\right)}\right] \\ \Rightarrow \frac{-3^{-y}}{\log \left(3\right)}=\frac{3^x}{\log \left(3\right)}+k \\ \Rightarrow \frac{-3^{-y}}{\log \left(3\right)}=\frac{3^x+k\log \left(3\right)}{\log \left(3\right)} \\ \Rightarrow -3^{-y}=3^x+k\log \left(3\right) \end{gathered}$$
$\Rightarrow 3^x+3^{-y}=c$, where $c=-k\log \left(3\right)$

Hence, option (D) is the correct answer.