https://file.aiquickdraw.com/mathbot/file/1726512364873zfnkdpg1.jpg

The equation in the image is:

\[
n \frac{dy}{dx} + n^2 y = e^x
\]

This is a first-order linear differential equation in terms of \( y \). We can solve this using the method of an integrating factor.

### Step 1: Rewrite the equation
First, divide the entire equation by \( n \), to simplify the expression:

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

### Step 2: Find the integrating factor
The integrating factor (I.F.) for a linear differential equation of the form \( \frac{dy}{dx} + P(x)y = Q(x) \) is:

\[
I.F. = e^{\int P(x) dx}
\]

Here, \( P(x) = n \), so:

\[
I.F. = e^{\int n \, dx} = e^{nx}
\]

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

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

Simplifying the right-hand side:

\[
\frac{e^{(n+1)x}}{n}
\]

The left-hand side becomes the derivative of \( y \cdot e^{nx} \):

\[
\frac{d}{dx} \left( y e^{nx} \right) = \frac{e^{(n+1)x}}{n}
\]

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

\[
y e^{nx} = \int \frac{e^{(n+1)x}}{n} \, dx
\]

The integral on the right-hand side is:

\[
\frac{e^{(n+1)x}}{n(n+1)} + C
\]

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

\[
y = \frac{e^{(n+1)x}}{n(n+1) e^{nx}} + \frac{C}{e^{nx}}
\]

Simplifying:

\[
y = \frac{e^x}{n(n+1)} + C e^{-nx}
\]

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

\[
y = \frac{e^x}{n(n+1)} + C e^{-nx}
\]

Let me know if you want more details or have any questions!

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Here are five related questions to expand on this topic:
1. What is the method of integrating factors, and why does it work for linear differential equations?
2. How do you verify the solution to a differential equation?
3. What would happen if the equation had a different right-hand side, such as \( e^{-x} \)?
4. Could you solve the equation using another method, such as variation of parameters?
5. What is the significance of the constant \( C \) in the general solution?

**Tip**: Always check the form of a differential equation to determine the best method for solving it. Some may be separable, while others require specific techniques like integrating factors.

Solve the differential equation: n dy/dx + n^2 y = e^x

Learn how to solve the first-order linear differential equation n dy/dx + n^2 y = e^x using the method of integrating factors. This problem covers core concepts in calculus, including the integrating factor and the process of solving linear differential equations. Suitable for college-level students.

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