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To solve the differential equation:

\[
\frac{dy}{dx} = 20x e^{5x^2}
\]

and find the particular solution satisfying \( y(0) = 2 \), let's proceed as follows:

### Step 1: Recognize the Structure of the Differential Equation
The equation given can be written in terms of a derivative of a function:
\[
\frac{dy}{dx} = 20x e^{5x^2}
\]
Notice that \(20x e^{5x^2}\) resembles the derivative of \(e^{5x^2}\) with respect to \(x\), since:
\[
\frac{d}{dx} e^{5x^2} = e^{5x^2} \cdot \frac{d}{dx}(5x^2) = 10x e^{5x^2}
\]
Therefore, multiplying by 2, we get:
\[
20x e^{5x^2} = 2 \cdot \frac{d}{dx}(e^{5x^2})
\]

### Step 2: Integrate Both Sides
Thus, the differential equation can be rewritten as:
\[
\frac{dy}{dx} = 2 \cdot \frac{d}{dx}(e^{5x^2})
\]
Integrating both sides with respect to \(x\):
\[
y = 2e^{5x^2} + C
\]
where \(C\) is the constant of integration.

### Step 3: Apply the Initial Condition \( y(0) = 2 \)
To find \(C\), use the initial condition \(y(0) = 2\):
\[
2 = 2e^{5 \cdot 0^2} + C
\]
\[
2 = 2 \cdot 1 + C
\]
\[
C = 0
\]

### Final Solution
Thus, the particular solution to the differential equation is:
\[
y = 2e^{5x^2}
\]

Would you like more details on any step, or do you have any questions?

### Related Questions
1. How do we verify if a function is a solution to a differential equation?
2. What are initial conditions, and why are they important in differential equations?
3. How does the method of integration by recognition work?
4. Can we solve this differential equation using separation of variables?
5. How would the solution change if the initial condition were different?

### Tip
In differential equations, recognizing terms as derivatives of known functions can simplify finding solutions efficiently.

Find the general solution of the differential equation \( \frac{dy}{dx} = 20x e^{5x^2} \) and give the particular solution satisfying \( y(0) = 2 \).

Learn how to find the general and particular solutions to the differential equation dy/dx = 20x e^(5x^2) with an initial condition y(0) = 2. This solution involves exponential functions and integration by recognition, suitable for undergraduate mathematics students.

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