# Solve the first-order ODE: $$\frac{dy}{dx} = 3y$$

1. Write down the given differential equation:

### $\frac{dy}{dx} = 3y$

2. Separate the variables:
To separate the variables, we need to get all the $$y$$-terms on one side and the $$x$$-terms on the other side. Divide both sides by $$y$$ (assuming $$y \neq 0$$) and multiply both sides by $$dx$$:
$\frac{1}{y} \, dy = 3 \, dx$

3. Integrate both sides:
Integrate the left side with respect to $$y$$ and the right side with respect to $$x$$:
$\int \frac{1}{y} \, dy = \int 3 \, dx$

4. Evaluate the integrals:
The integral of $$\frac{1}{y}$$ with respect to $$y$$ is $$\ln|y|$$, and the integral of $$3$$ with respect to $$x$$ is $$3x$$:
$\ln|y| = 3x + C$
Here, $$C$$ is the constant of integration.

5. Solve for $$y$$:
To solve for $$y$$, we exponentiate both sides to remove the natural logarithm:
$e^{\ln|y|} = e^{3x + C}$
Since $$e^{\ln|y|} = |y|$$ and $$e^{3x + C} = e^{3x} \cdot e^C$$, we get:
$|y| = e^{3x} \cdot e^C$
Let $$e^C = C’$$, where $$C’$$ is a new constant (still a constant, but written in a simpler form):
$|y| = C’ e^{3x}$
Since $$C’$$ can be any positive constant, we write it as a general constant $$C$$ which can be positive or negative:
$y = Ce^{3x}$
where $$C$$ is an arbitrary constant.

The general solution to the differential equation $$\frac{dy}{dx} = 3y$$ is:
### $y = Ce^{3x}$
where $$C$$ is an arbitrary constant.