The given differential equation is:

$\frac{dP}{dt} + 2tP = P - 2 + 4t$

Step 1: Rewrite the equation

First, let's move all terms involving $P$ to the left side and constant or independent terms to the right side:

$\frac{dP}{dt} + 2tP - P = 4t - 2$

Simplify the terms involving $P$:

$\frac{dP}{dt} + (2t - 1)P = 4t - 2$

This is a first-order linear differential equation of the form $\frac{dP}{dt} + a(t)P = b(t)$, where $a(t) = 2t - 1$ and $b(t) = 4t - 2$.

Step 2: Find the integrating factor

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

$\mu(t) = e^{\int a(t) \, dt} = e^{\int (2t - 1) \, dt}$

Let's calculate this integral:

$\mu(t) = e^{t^2 - t}$

Step 3: Multiply the differential equation by the integrating factor

We multiply both sides of the differential equation by $e^{t^2 - t}$:

$e^{t^2 - t} \frac{dP}{dt} + (2t - 1)e^{t^2 - t}P = (4t - 2)e^{t^2 - t}$

The left-hand side is the derivative of $\mu(t)P$:

$\frac{d}{dt}\left(e^{t^2 - t}P\right) = (4t - 2)e^{t^2 - t}$

Step 4: Integrate both sides with respect to $t$

Integrate both sides:

$e^{t^2 - t}P = \int (4t - 2)e^{t^2 - t} \, dt$

Step 5: Solve the integral

Let's solve the integral on the right-hand side using substitution. Set $u = t^2 - t$, then $du = (2t - 1) dt$.

This means we can rewrite the integral as:

$\int (4t - 2)e^{t^2 - t} \, dt = 2 \int e^u \, du = 2e^u + C = 2e^{t^2 - t} + C$

Step 6: Solve for $P(t)$

Finally, solve for $P(t)$:

$e^{t^2 - t}P = 2e^{t^2 - t} + C$

Dividing by $e^{t^2 - t}$ on both sides:

$P(t) = 2 + Ce^{-(t^2 - t)}$

Final Solution:

The general solution to the differential equation is:

$P(t) = 2 + Ce^{-(t^2 - t)}$

where $C$ is the constant of integration.

Would you like more details or have any other questions? Here are some related questions to consider:

1. What is the significance of the constant $C$ in the solution?
2. How does the solution behave as $t$ becomes very large or very small?
3. Can you solve for $C$ if an initial condition $P(t_0) = P_0$ is given?
4. How does the form of $a(t)$ in the integrating factor affect the solution?
5. How would the solution change if the non-homogeneous term were different?

Tip: After finding the general solution to a differential equation, always check if it satisfies the original equation by substituting it back in.