The given equation is:

$x \frac{dy}{dx} + (1 + x)y = x^3$

This is a first-order linear differential equation, which can be solved using the integrating factor method.

Step 1: Rewrite the equation in standard form

$\frac{dy}{dx} + \frac{(1 + x)}{x}y = x^2$

Here:

• $P(x) = \frac{1 + x}{x}$,
• $Q(x) = x^2$.

Step 2: Compute the integrating factor

The integrating factor $I(x)$ is given by: $I(x) = e^{\int P(x) \, dx} = e^{\int \frac{1 + x}{x} \, dx}$

Break the integral: $\int \frac{1 + x}{x} \, dx = \int \frac{1}{x} \, dx + \int 1 \, dx = \ln|x| + x$

Thus: $I(x) = e^{\ln|x| + x} = e^{\ln|x|} \cdot e^x = x e^x$

Step 3: Multiply through by the integrating factor

Multiply the entire equation by $I(x) = x e^x$: $x e^x \frac{dy}{dx} + x e^x \cdot \frac{1 + x}{x} y = x^3 e^x$

Simplify: $x e^x \frac{dy}{dx} + (1 + x)e^x y = x^3 e^x$

Notice the left-hand side is the derivative of $(e^x x y)$: $\frac{d}{dx} (e^x x y) = x^3 e^x$

Step 4: Integrate both sides

Integrate with respect to $x$: $e^x x y = \int x^3 e^x \, dx$

To integrate $\int x^3 e^x \, dx$, use integration by parts repeatedly. Let:

• $u = x^3$, $dv = e^x dx$,
• $du = 3x^2 dx$, $v = e^x$.

Using the formula $\int u \, dv = uv - \int v \, du$: $\int x^3 e^x \, dx = x^3 e^x - \int 3x^2 e^x \, dx$

Repeat this process for $\int x^2 e^x dx$ and $\int x e^x dx$. After completing all steps, the result is: $\int x^3 e^x \, dx = e^x (x^3 - 3x^2 + 6x - 6)$

Thus: $e^x x y = e^x (x^3 - 3x^2 + 6x - 6) + C$

Step 5: Solve for $y$

Divide through by $e^x x$: $y = \frac{x^3 - 3x^2 + 6x - 6}{x} + \frac{C}{x e^x}$

Simplify: $y = x^2 - 3x + 6 - \frac{6}{x} + \frac{C}{x e^x}$

Final Solution:

$y = x^2 - 3x + 6 - \frac{6}{x} + \frac{C}{x e^x}$

Let me know if you'd like detailed steps on any part of the process!

Questions:

1. What is the significance of an integrating factor in solving linear differential equations?
2. How do you calculate an integrating factor for first-order equations?
3. What are the steps to simplify after finding the integrating factor?
4. Why do we use integration by parts for certain terms during integration?
5. How does the constant of integration $C$ affect the final solution?

Tip: Always rewrite differential equations in their standard form before applying solution methods like the