The given differential equation is:

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

This is a first-order linear differential equation of the form:

$\frac{dy}{dx} + P(x)y = Q(x)$

where $P(x) = \frac{1}{x}$ and $Q(x) = x^3$.

Step 1: Find the Integrating Factor (IF)

The integrating factor $\mu(x)$ is given by:

$\mu(x) = e^{\int P(x) dx} = e^{\int \frac{1}{x} dx} = e^{\ln|x|} = |x|$

So, the integrating factor is $\mu(x) = x$ (we can drop the absolute value if we assume $x > 0$).

Step 2: Multiply the whole equation by the integrating factor

Multiplying both sides of the differential equation by $x$:

$x \frac{dy}{dx} + y = x^4$

Step 3: Solve the equation

The left-hand side of the equation is now the derivative of $yx$:

$\frac{d}{dx}(yx) = x^4$

Integrating both sides with respect to $x$:

$yx = \int x^4 dx = \frac{x^5}{5} + C$

where $C$ is the constant of integration.

Step 4: Solve for $y$

Finally, divide by $x$ to isolate $y$:

$y = \frac{x^4}{5} + \frac{C}{x}$

Thus, the solution to the differential equation is:

$y = \frac{x^4}{5} + \frac{C}{x}$

Do you want any further details or have any questions?

Here are 5 questions related to this topic:

1. What are some examples of first-order linear differential equations?
2. How is an integrating factor used in solving linear differential equations?
3. How do we solve differential equations with non-constant coefficients?
4. What is the significance of the constant of integration?
5. Can a linear differential equation always be solved using an integrating factor?

Tip: Always check if a first-order linear differential equation can be simplified using an integrating factor. It makes solving such equations straightforward.