To solve the differential equation \(\frac{dy}{dx} = 2xy + 3x^2 e^{x^2}\) with the initial condition \(y(0) = 5\), we can use an integrating factor method or attempt to solve it as a first-order linear differential equation.

### Step 1: Rearrange the equation
The equation can be written in the form:

\[
\frac{dy}{dx} - 2xy = 3x^2 e^{x^2}
\]

This is a linear differential equation of the form:

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

where \(P(x) = -2x\) and \(Q(x) = 3x^2 e^{x^2}\).

### Step 2: Find the integrating factor
The integrating factor \( \mu(x) \) is given by:

\[
\mu(x) = e^{\int P(x) dx} = e^{\int -2x dx} = e^{-x^2}
\]

### Step 3: Multiply the differential equation by the integrating factor
Multiply the entire differential equation by \( \mu(x) = e^{-x^2} \):

\[
e^{-x^2} \frac{dy}{dx} - 2xe^{-x^2}y = 3x^2 e^{x^2} e^{-x^2}
\]

which simplifies to:

\[
e^{-x^2} \frac{dy}{dx} - 2xe^{-x^2}y = 3x^2
\]

The left-hand side can be recognized as the derivative of \( y \cdot e^{-x^2} \):

\[
\frac{d}{dx}\left( y \cdot e^{-x^2} \right) = 3x^2
\]

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

\[
y \cdot e^{-x^2} = \int 3x^2 dx
\]

The integral on the right-hand side is:

\[
\int 3x^2 dx = x^3 + C
\]

So, the equation becomes:

\[
y \cdot e^{-x^2} = x^3 + C
\]

### Step 5: Solve for \( y \)
Multiply both sides by \( e^{x^2} \):

\[
y = e^{x^2} \left( x^3 + C \right)
\]

### Step 6: Apply the initial condition
Use the initial condition \( y(0) = 5 \) to find the constant \(C\):

\[
5 = e^{0} \cdot (0^3 + C)
\]

\[
C = 5
\]

### Final Solution
The solution to the differential equation is:

\[
y = e^{x^2} \left( x^3 + 5 \right)
\]

Would you like more details or have any questions about this solution?

Here are 5 related questions for further exploration:
1. How would the solution change if the initial condition were \( y(1) = 5 \) instead of \( y(0) = 5 \)?
2. What is the general solution for \(\frac{dy}{dx} = 2xy + 3x^2 e^{x^2}\) without any initial condition?
3. How would you find the particular solution if \( P(x) \) were not linear in \(x\)?
4. Can this method be applied to non-linear differential equations?
5. How would the solution be affected if the function \( Q(x) \) had a different form?

**Tip:** Always check your integrating factor calculation, as it is crucial for simplifying the equation correctly.

Learn how to solve the differential equation dy/dx = 2xy + 3x²e^x² with the initial condition y(0) = 5. Detailed step-by-step solution using the integrating factor method and linear differential equation techniques. Suitable for undergraduate-level students and those interested in differential equations.

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