Find the general solution to the first order linear differential equation
dy/dt + 3y = sin(3t)

The given first-order linear differential equation is:

\[
\frac{dy}{dt} + 3y = \sin(3t)
\]

### Step 1: Identify the standard form
This is a linear differential equation of the form:

\[
\frac{dy}{dt} + P(t) y = Q(t)
\]

Where \( P(t) = 3 \) and \( Q(t) = \sin(3t) \).

### Step 2: Find the integrating factor
The integrating factor \( \mu(t) \) is given by:

\[
\mu(t) = e^{\int P(t) dt} = e^{\int 3 dt} = e^{3t}
\]

### Step 3: Multiply the equation by the integrating factor
Multiply both sides of the differential equation by \( e^{3t} \):

\[
e^{3t} \frac{dy}{dt} + 3e^{3t} y = e^{3t} \sin(3t)
\]

### Step 4: Simplify the left-hand side
The left-hand side becomes the derivative of \( e^{3t} y \):

\[
\frac{d}{dt} \left( e^{3t} y \right) = e^{3t} \sin(3t)
\]

### Step 5: Integrate both sides
Now, integrate both sides with respect to \( t \):

\[
e^{3t} y = \int e^{3t} \sin(3t) dt
\]

The integral on the right-hand side can be solved using integration by parts or a known result:

\[
\int e^{3t} \sin(3t) dt = \frac{e^{3t} \left( \sin(3t) - 3 \cos(3t) \right)}{18} + C
\]

### Step 6: Solve for \( y(t) \)
Now, divide both sides by \( e^{3t} \) to solve for \( y(t) \):

\[
y(t) = \frac{\sin(3t) - 3 \cos(3t)}{18} + Ce^{-3t}
\]

### General Solution:
Thus, the general solution to the differential equation is:

\[
y(t) = \frac{\sin(3t) - 3 \cos(3t)}{18} + Ce^{-3t}
\]

Where \( C \) is the constant of integration.

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Would you like further details on any step? Here are some related questions:

1. How do you compute the integrating factor for a linear differential equation?
2. What is the method of integrating by parts, and how is it used here?
3. How do you determine the constant of integration \( C \)?
4. Can this solution be verified by substitution into the original equation?
5. What are some typical applications of first-order linear differential equations?

**Tip:** The integrating factor method is very useful for solving linear first-order differential equations, so it's important to master it!

Find the general solution to the first order linear differential equation dy/dt + 3y = sin(3t)

Explore the detailed solution to the first-order linear differential equation dy/dt + 3y = sin(3t) using the integrating factor method. This guide walks through the step-by-step process, including the identification of the integrating factor, solving for y(t), and understanding the general solution. Ideal for college-level students.

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