Solve the differential equation: $\frac{\mathrm{dy}}{\mathrm{dx}}+2\mathrm{x}={\mathrm{e}}^{3\mathrm{x}}$

Given:  $\frac{\mathrm{dy}}{\mathrm{dx}}+2\mathrm{x}={\mathrm{e}}^{3\mathrm{x}}$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}={\mathrm{e}}^{3\mathrm{x}}-2\mathrm{x}$

$\Rightarrow \mathrm{dy}=\left({\mathrm{e}}^{3\mathrm{x}}-2\mathrm{x}\right)\mathrm{dx}$

Integrating both sides with respect x, we get

$$\begin{gathered} \Rightarrow \int \mathrm{dy}=\int \left({\mathrm{e}}^{3\mathrm{x}}-2\mathrm{x}\right)\mathrm{dx} \\ \Rightarrow \mathrm{y}=\frac{{\mathrm{e}}^{3\mathrm{x}}}{3}-2\frac{{\mathrm{x}}^2}{2}+\mathrm{C} \\ \Rightarrow \mathrm{y}=\frac{{\mathrm{e}}^{3\mathrm{x}}}{3}-{\mathrm{x}}^2+\mathrm{C} \end{gathered}$$

$\Rightarrow y+x^2=\frac{e^{3x}}{3}+C$

So, it is defined for all $\mathrm{x}\in \mathrm{R}$

Hence, $\Rightarrow \mathrm{y}+{\mathrm{x}}^2=\frac{{\mathrm{e}}^{3\mathrm{x}}}{3}+\mathrm{C}$, where $\mathrm{x}\in \mathrm{R}$ is the solution to the given differential equation.

Which is the required answer. x∈R