The given differential equation is:

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

This is a first-order linear differential equation. To solve it, we will use an integrating factor.

### Step 1: Rewrite the equation in standard form
The equation can be rewritten as:

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

This is now in the form \(\frac{dy}{dx} + P(x)y = Q(x)\), where \(P(x) = 3\) and \(Q(x) = e^x\).

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

\[
\mu(x) = e^{\int P(x) dx} = e^{\int 3 dx} = e^{3x}
\]

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

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

This simplifies to:

\[
\frac{d}{dx} \left( e^{3x} y \right) = e^{4x}
\]

### Step 4: Integrate both sides
Now, integrate both sides with respect to \(x\):

\[
\int \frac{d}{dx} \left( e^{3x} y \right) dx = \int e^{4x} dx
\]

On the left side, the integral is simply \( e^{3x} y \), and on the right side, we get:

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

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

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

Simplifying this:

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

Thus, the general solution to the differential equation is:

\[
y(x) = \frac{e^x}{4} + C e^{-3x}
\]

### Do you want more details or have any questions?

Here are five related questions:
1. How can we determine the constant \(C\) if an initial condition is provided?
2. What are some typical applications of first-order linear differential equations?
3. How do integrating factors simplify solving linear differential equations?
4. Can this method be applied to non-homogeneous linear differential equations?
5. What is the behavior of the solution as \(x \to \infty\)?

**Tip:** When solving linear differential equations, always start by expressing the equation in the standard form \(\frac{dy}{dx} + P(x)y = Q(x)\) before finding the integrating factor.

Learn how to solve the first-order linear differential equation dy/dx = e^x - 3y using the integrating factor method. This problem involves key calculus concepts such as differential equations and integrating factors. Suitable for undergraduate students studying Calculus I.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation