The given initial value problem is:

$\frac{dy}{dt} + 2y = 6, \quad y(0) = 0$

This is a first-order linear non-homogeneous differential equation. To solve it, we will use the method of integrating factors.

Step 1: Identify the components

The equation is of the form:

$\frac{dy}{dt} + P(t)y = Q(t)$

where $P(t) = 2$ and $Q(t) = 6$.

Step 2: Compute the integrating factor

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

$\mu(t) = e^{\int P(t) \, dt} = e^{\int 2 \, dt} = e^{2t}$

Step 3: Multiply the entire equation by the integrating factor

Multiply both sides of the differential equation by $e^{2t}$:

$e^{2t} \frac{dy}{dt} + 2 e^{2t} y = 6 e^{2t}$

The left side of this equation is the derivative of $e^{2t} y$, so we rewrite the equation as:

$\frac{d}{dt} \left( e^{2t} y \right) = 6 e^{2t}$

Step 4: Integrate both sides

Now, integrate both sides with respect to $t$:

$\int \frac{d}{dt} \left( e^{2t} y \right) \, dt = \int 6 e^{2t} \, dt$

This gives:

$e^{2t} y = 3 e^{2t} + C$

where $C$ is the constant of integration.

Step 5: Solve for $y(t)$

Now, solve for $y(t)$:

$y(t) = 3 + C e^{-2t}$

Step 6: Apply the initial condition

We are given the initial condition $y(0) = 0$. Substituting $t = 0$ and $y(0) = 0$ into the equation:

$0 = 3 + C e^{0}$

$0 = 3 + C$

$C = -3$

Step 7: Write the particular solution

Thus, the particular solution to the differential equation is:

$y(t) = 3 - 3 e^{-2t}$

Final Answer:

The particular solution to the initial value problem is:

$y(t) = 3 - 3 e^{-2t}$

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Would you like more details on any part of the solution? Here are some related questions to expand the topic:

1. How can we solve a non-homogeneous differential equation using the method of undetermined coefficients?
2. What happens if the initial condition were different (e.g., $y(0) = 1$)?
3. How do you identify whether a differential equation is homogeneous or non-homogeneous?
4. How does the choice of the integrating factor affect the solution process?
5. Can the method of integrating factors be applied to second-order differential equations?

Tip: For a first-order linear differential equation, the method of integrating factors is usually the simplest approach, but always make sure to carefully compute the integrating factor!