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The given differential equation is:

\[
\frac{dp}{dt} = t^2 p - p + t^2 - L
\]

This is a first-order, linear non-homogeneous differential equation in \( p(t) \). To solve it, we need to find the general solution.

Here's the step-by-step approach to solve it:

### Step 1: Rearrange the equation
Group the terms involving \( p \) and terms not involving \( p \):

\[
\frac{dp}{dt} + (1 - t^2)p = t^2 - L
\]

### Step 2: Find the integrating factor
The equation is in the standard linear form:

\[
\frac{dp}{dt} + P(t)p = Q(t)
\]

where \( P(t) = 1 - t^2 \) and \( Q(t) = t^2 - L \).

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

\[
\mu(t) = e^{\int P(t) dt} = e^{\int (1 - t^2) dt} = e^{t - \frac{t^3}{3}}
\]

### Step 3: Multiply the equation by the integrating factor
Multiply both sides of the equation by \( \mu(t) = e^{t - \frac{t^3}{3}} \) to simplify:

\[
e^{t - \frac{t^3}{3}} \frac{dp}{dt} + e^{t - \frac{t^3}{3}}(1 - t^2)p = (t^2 - L)e^{t - \frac{t^3}{3}}
\]

The left-hand side becomes:

\[
\frac{d}{dt}\left( e^{t - \frac{t^3}{3}} p \right)
\]

So the equation is:

\[
\frac{d}{dt} \left( e^{t - \frac{t^3}{3}} p \right) = (t^2 - L)e^{t - \frac{t^3}{3}}
\]

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

\[
e^{t - \frac{t^3}{3}} p = \int (t^2 - L)e^{t - \frac{t^3}{3}} dt + C
\]

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

To fully solve the integral on the right-hand side, more information may be needed, or numerical methods can be used depending on the value of \( L \). However, this is the general framework for solving this type of first-order linear differential equation.

---

Would you like further clarification or details on any step?

Here are some related questions for deeper exploration:
1. How do you solve an integral involving exponential functions with polynomials in the exponent?
2. What are the general methods to solve linear non-homogeneous differential equations?
3. How does the integrating factor method simplify solving first-order linear ODEs?
4. Can this equation be solved numerically if \( L \) is a constant or function of \( t \)?
5. How can the solution be visualized for specific values of \( L \)?

**Tip**: Always check if the differential equation can be written in a standard form before applying methods like the integrating factor!

Solve the differential equation: dp/dt = t^2 * p - p + t^2 - L

Learn how to solve the first-order linear non-homogeneous differential equation dp/dt = t^2 * p - p + t^2 - L using the integrating factor method. This step-by-step solution covers key mathematical concepts, including the use of an integrating factor and linear differential equations, suitable for undergraduate students.

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