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

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

A: $$y = Ce^{3x}$$

### Worksheet

Step 1: Recognize the Form
The given differential equation is of the form $$\frac{dy}{dx} = ky$$, where $$k$$ is a constant (in this case, $$k = 3$$). This is a separable differential equation.

Step 2: Separate the Variables
To separate the variables, we rewrite the equation so that all terms involving $$y$$ are on one side and all terms involving $$x$$ are on the other side:
$\frac{dy}{y} = 3 \, dx$

Step 3: Integrate Both Sides
Integrate both sides of the equation:
$\int \frac{1}{y} \, dy = \int 3 \, dx$

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

Step 5: Solve for $$y$$
To solve for $$y$$, we exponentiate both sides to get rid of the natural logarithm:
$e^{\ln|y|} = e^{3x + C}$

Since $$e^{\ln|y|} = |y|$$, we can write:
$|y| = e^{3x + C}$

We can simplify $$e^{3x + C}$$ as $$e^{3x} \cdot e^C$$. Let $$e^C = C'$$, where $$C'$$ is a new constant (which can be positive or negative). Thus:
$|y| = C' e^{3x}$

Therefore, $$y = \pm C' e^{3x}$$. To simplify the notation, we can combine the $$\pm$$ and $$C'$$ into a single constant $$C$$, which can be any real number. Hence:
$y = C e^{3x}$

Final Solution
The general solution to the differential equation $$\frac{dy}{dx} = 3y$$ is:
$y = C e^{3x}$

where $$C$$ is an arbitrary constant.