To solve the differential equation $\frac{dy}{dx} + xy = x^3$, we can use the method of integrating factors. The equation is in the standard linear form:

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

where $P(x) = x$ and $Q(x) = x^3$.

Step 1: Find the integrating factor

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

$\mu(x) = e^{\int P(x) \, dx} = e^{\int x \, dx} = e^{\frac{x^2}{2}}$

Step 2: Multiply the entire equation by the integrating factor

Multiply the equation by $e^{\frac{x^2}{2}}$:

$e^{\frac{x^2}{2}} \frac{dy}{dx} + e^{\frac{x^2}{2}} xy = e^{\frac{x^2}{2}} x^3$

Step 3: Simplify the left side

The left side can be expressed as the derivative of a product:

$\frac{d}{dx}\left( e^{\frac{x^2}{2}} y \right) = e^{\frac{x^2}{2}} x^3$

Step 4: Integrate both sides

Integrate both sides with respect to $x$:

$\int \frac{d}{dx}\left( e^{\frac{x^2}{2}} y \right) \, dx = \int e^{\frac{x^2}{2}} x^3 \, dx$

The left side simplifies to:

$e^{\frac{x^2}{2}} y = \int e^{\frac{x^2}{2}} x^3 \, dx + C$

Step 5: Solve the right-hand integral

The integral $\int e^{\frac{x^2}{2}} x^3 \, dx$ does not have a standard elementary form and may need to be evaluated using numerical methods or special functions, depending on the context.

Step 6: Solve for $y$

Once the integral is computed, we can express $y$:

$y = e^{-\frac{x^2}{2}} \left( \int e^{\frac{x^2}{2}} x^3 \, dx + C \right)$

Summary

The general solution to the differential equation is:

$y = e^{-\frac{x^2}{2}} \left( \int e^{\frac{x^2}{2}} x^3 \, dx + C \right)$

Would you like to explore the integral further or have any other questions? Here are some related topics to consider:

1. What are the properties of linear differential equations?
2. How do you compute integrals of exponential functions?
3. What are some applications of this type of differential equation?
4. Can you provide examples of similar equations?
5. What techniques can be used to solve non-linear differential equations?

Tip: Familiarize yourself with integrating factors as they are a powerful tool for solving linear differential equations.