Solve a differential equation for example 

Sure! Let's solve a differential equation. I'll start with a common example:

### Example: Solve the differential equation
\[
\frac{dy}{dx} + y = e^x.
\]

This is a first-order linear differential equation. 

#### Step-by-Step Solution:

1. **Identify the standard form:**  
   The standard form of a first-order linear differential equation is:
   \[
   \frac{dy}{dx} + P(x)y = Q(x),
   \]
   where \( P(x) = 1 \) and \( Q(x) = e^x \) in our case.

2. **Find the integrating factor (IF):**  
   The integrating factor is given by:
   \[
   IF = e^{\int P(x) \, dx} = e^{\int 1 \, dx} = e^x.
   \]

3. **Multiply the entire differential equation by the integrating factor:**  
   Multiply both sides of the equation by \( e^x \):
   \[
   e^x \frac{dy}{dx} + e^x y = e^x \cdot e^x.
   \]
   This simplifies to:
   \[
   e^x \frac{dy}{dx} + e^x y = e^{2x}.
   \]

4. **Recognize the left side as the derivative of a product:**  
   Notice that the left-hand side is the derivative of \( e^x y \):
   \[
   \frac{d}{dx} (e^x y) = e^{2x}.
   \]

5. **Integrate both sides:**  
   Integrate with respect to \( x \):
   \[
   e^x y = \int e^{2x} \, dx.
   \]

   The integral on the right is:
   \[
   \int e^{2x} \, dx = \frac{e^{2x}}{2} + C,
   \]
   where \( C \) is the constant of integration.

6. **Solve for \( y \):**  
   Now, solve for \( y \):
   \[
   e^x y = \frac{e^{2x}}{2} + C \implies y = \frac{e^{2x}}{2e^x} + \frac{C}{e^x}.
   \]

   Simplify further:
   \[
   y = \frac{e^x}{2} + Ce^{-x}.
   \]

### Final Solution:
\[
y(x) = \frac{e^x}{2} + Ce^{-x}.
\]

This is the general solution to the differential equation.

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Would you like more details on this solution, or do you have any questions?

### 5 Related Questions:
1. How do you solve a second-order differential equation?
2. What are homogeneous differential equations and how do you solve them?
3. Can you explain the method of separation of variables for differential equations?
4. How do you solve differential equations using Laplace transforms?
5. What is the significance of initial conditions in solving differential equations?

### Tip:
When solving differential equations, always check if the equation can be simplified or rewritten in a standard form to identify the correct solving method easily!

Solve a differential equation for example

This page provides a detailed step-by-step solution to the differential equation dy/dx + y = e^x. Learn how to apply the integrating factor method and solve for the general solution. Suitable for undergraduate students studying calculus or differential equations.

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