The given differential equation is:

\[
\frac{dy}{dx} + y = e^{3x}
\]

This is a first-order linear differential equation of the form:

\[
\frac{dy}{dx} + P(x) y = Q(x)
\]

where \( P(x) = 1 \) and \( Q(x) = e^{3x} \).

To solve this, we can use the method of integrating factors.

### Step 1: Find the Integrating Factor (IF)

The integrating factor \( \mu(x) \) is given by:

\[
\mu(x) = e^{\int P(x) \, dx} = e^{\int 1 \, dx} = e^x
\]

### Step 2: Multiply the Equation by the Integrating Factor

Multiply both sides of the differential equation by the integrating factor \( e^x \):

\[
e^x \frac{dy}{dx} + e^x y = e^x e^{3x}
\]

Simplify:

\[
e^x \frac{dy}{dx} + e^x y = e^{4x}
\]

### Step 3: Rewrite the Left Side as a Derivative

Notice that the left side is the derivative of \( e^x y \):

\[
\frac{d}{dx} \left( e^x y \right) = e^{4x}
\]

### Step 4: Integrate Both Sides

Integrate both sides with respect to \( x \):

\[
e^x y = \int e^{4x} \, dx
\]

Compute the integral on the right side:

\[
e^x y = \frac{e^{4x}}{4} + C
\]

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

### Step 5: Solve for \( y \)

Now solve for \( y \) by dividing both sides by \( e^x \):

\[
y = \frac{e^{4x}}{4e^x} + \frac{C}{e^x}
\]

Simplify the expression:

\[
y = \frac{e^{3x}}{4} + Ce^{-x}
\]

### Final Solution

The general solution to the differential equation is:

\[
\boxed{y = \frac{e^{3x}}{4} + Ce^{-x}}
\]

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

---

Would you like more details or have any questions about the solution?

### Here are 5 related questions to explore further:
1. How do you solve a differential equation using an integrating factor?
2. What is the significance of a linear differential equation being first-order?
3. How do you determine the particular solution if an initial condition is given?
4. What are other methods to solve first-order linear differential equations?
5. How can you check if your solution to a differential equation is correct?

**Tip:** Always verify the solution by differentiating it and substituting it back into the original equation to ensure it satisfies the equation.

Learn how to solve the first-order linear differential equation dy/dx + y = e^3x using the method of integrating factors. This solution involves key calculus concepts, including finding the integrating factor and solving the differential equation step-by-step. Suitable for college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation