The given differential equation is:

\[
\frac{dP}{dt} + 2tP = P + 4t^{-2}
\]

To solve this first-order linear differential equation, let's rewrite it in standard form:

\[
\frac{dP}{dt} - P + 2tP = 4t^{-2}
\]

Combining like terms:

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

This is a 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 1: 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:

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

### Step 2: Multiply the entire differential equation by the integrating factor
We multiply both sides of the 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 3: Integrate both sides with respect to \(t\)
Integrating both sides:

\[
e^{t^2 - t}P = \int 4t^{-2}e^{t^2 - t} \, dt
\]

This integral on the right-hand side is non-trivial, so solving it exactly would require special functions or numerical methods. For now, let's represent the solution as:

\[
P(t) = e^{-(t^2 - t)}\left(C + \int 4t^{-2}e^{t^2 - t} \, dt\right)
\]

where \(C\) is the constant of integration.

### Summary:
The solution to the differential equation is:

\[
P(t) = e^{-(t^2 - t)}\left(C + \int 4t^{-2}e^{t^2 - t} \, dt\right)
\]

This expression gives the general solution, though the integral may need to be evaluated using numerical or approximate methods.

Would you like to explore the integral further or have any other questions? Here are some related questions to consider:

1. How can the integral \(\int 4t^{-2}e^{t^2 - t} \, dt\) be evaluated numerically?
2. What is the physical interpretation of the integrating factor in solving linear differential equations?
3. How does the method of separation of variables differ from the method used here?
4. Can you derive a particular solution given an initial condition for \(P(t)\)?
5. How would the solution change if the equation were non-linear?

**Tip:** Always check the form of your differential equation before applying a solution method to ensure you're using the correct approach.

Learn how to solve the first-order linear differential equation dP/dt + 2tP = P + 4t^{-2}. This problem involves finding the integrating factor and integrating the equation step-by-step. Suitable for undergraduate students studying differential equations.

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