To solve the differential equation $y' + 4xy = 8x$ using the method of integrating factors, let's go through the steps carefully:

Step 1: Rewrite the equation in standard form

The given differential equation is:

$y' + 4xy = 8x$

We can rewrite this in the standard linear form $y' + P(x)y = Q(x)$, where $P(x) = 4x$ and $Q(x) = 8x$.

$y' + 4xy = 8x$

Step 2: Find the integrating factor

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

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

Substituting $P(x) = 4x$:

$\mu(x) = e^{\int 4x\, dx} = e^{2x^2}$

Step 3: Multiply the entire differential equation by the integrating factor

Now, multiply through the equation by the integrating factor $e^{2x^2}$:

$e^{2x^2} \cdot y' + e^{2x^2} \cdot 4xy = e^{2x^2} \cdot 8x$

This simplifies to:

$\frac{d}{dx} \left( y e^{2x^2} \right) = 8x e^{2x^2}$

Step 4: Integrate both sides

Now integrate both sides with respect to $x$:

$\int \frac{d}{dx} \left( y e^{2x^2} \right) \, dx = \int 8x e^{2x^2} \, dx$

The left-hand side simplifies directly to:

$y e^{2x^2}$

For the right-hand side, we use the substitution $u = 2x^2$, so $du = 4x\, dx$, and the integral becomes:

$\int 8x e^{2x^2} \, dx = 2 \int e^u \, du = 2e^u = 2e^{2x^2}$

Thus, we have:

$y e^{2x^2} = 2e^{2x^2} + C$

Step 5: Solve for $y$

Now solve for $y$:

$y = 2 + Ce^{-2x^2}$

Step 6: Verify the solution

To verify the solution, differentiate $y = 2 + Ce^{-2x^2}$ with respect to $x$:

$y' = -4xCe^{-2x^2}$

Substitute $y$ and $y'$ into the original equation $y' + 4xy = 8x$:

$-4xCe^{-2x^2} + 4x(2 + Ce^{-2x^2}) = 8x$

Simplifying:

$-4xCe^{-2x^2} + 8x + 4xCe^{-2x^2} = 8x$

The terms involving $Ce^{-2x^2}$ cancel out, leaving:

$8x = 8x$

This confirms that the solution is correct.

Final Solution:

$y = 2 + Ce^{-2x^2}$

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Would you like to see any additional details or clarifications on specific steps? Here are 5 related questions you can explore:

1. How do we find the general solution for non-homogeneous linear differential equations?
2. What is the role of the integrating factor in solving such differential equations?
3. Can the integrating factor method be applied to non-linear differential equations?
4. How can we solve a first-order differential equation using a substitution method instead of the integrating factor method?
5. What would happen if the right-hand side of the equation were a different function, like $6x^2$?

Tip: Always ensure the equation is in standard linear form $y' + P(x)y = Q(x)$ before applying the method of integrating factors.