We are given the first-order linear differential equation:

\[
\frac{dy}{dx} - \frac{y}{x} = x e^x
\]
with the initial condition \( y(1) = e - 1 \).

### Step 1: Rewrite in standard linear form
The given equation can be written in the form:
\[
\frac{dy}{dx} + P(x) y = Q(x)
\]
where \( P(x) = -\frac{1}{x} \) and \( Q(x) = x e^x \).

### Step 2: Find the integrating factor
The integrating factor \( \mu(x) \) is given by:
\[
\mu(x) = e^{\int P(x) \, dx} = e^{\int -\frac{1}{x} \, dx} = e^{-\ln|x|} = \frac{1}{x}
\]

### Step 3: Multiply through by the integrating factor
Multiply both sides of the differential equation by \( \frac{1}{x} \):
\[
\frac{1}{x} \frac{dy}{dx} - \frac{1}{x^2} y = e^x
\]
This simplifies to:
\[
\frac{d}{dx} \left( \frac{y}{x} \right) = e^x
\]

### Step 4: Integrate both sides
Integrate both sides with respect to \( x \):
\[
\frac{y}{x} = \int e^x \, dx = e^x + C
\]
where \( C \) is the constant of integration.

### Step 5: Solve for \( y \)
Multiply both sides by \( x \):
\[
y = x(e^x + C)
\]

### Step 6: Apply the initial condition \( y(1) = e - 1 \)
Substitute \( x = 1 \) and \( y = e - 1 \) into the equation:
\[
e - 1 = 1(e^1 + C) = e + C
\]
This simplifies to:
\[
C = -1
\]

### Step 7: Write the final solution
Substitute \( C = -1 \) into the general solution:
\[
y = x(e^x - 1)
\]

Thus, the solution to the differential equation is:
\[
y = x(e^x - 1)
\]

Would you like further details on any of these steps, or do you have any questions?

Here are some related questions for practice:
1. Solve the equation \(\frac{dy}{dx} + y/x = x^2 e^x\) with \( y(1) = 0 \).
2. What is the general solution for the differential equation \( \frac{dy}{dx} - y/x = 0 \)?
3. Solve \(\frac{dy}{dx} + 2y = x e^x\).
4. Verify the integrating factor method for \( \frac{dy}{dx} + y = x e^{2x} \).
5. How do initial conditions affect the solution to a differential equation?

**Tip:** For first-order linear differential equations, always look for the integrating factor to simplify the equation.

Learn how to solve the first-order linear differential equation dy/dx - y/x = xe^x with the initial condition y(1) = e - 1. This detailed step-by-step guide covers the integrating factor method, finding the general solution, and applying initial conditions. Suitable for advanced high school and 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