The given differential equation is:

$\frac{dP}{dt} = 4P + a$

where $a$ is a non-zero constant. We need to solve this differential equation for $P$, and include the constant of integration $C$.

Step 1: Rewrite the equation

$\frac{dP}{dt} = 4P + a$

This is a first-order linear differential equation. To solve it, let's first rewrite it as:

$\frac{dP}{dt} - 4P = a$

Step 2: Find the integrating factor

The standard form of a linear differential equation is:

$\frac{dP}{dt} + p(t)P = q(t)$

In our case, $p(t) = -4$ and $q(t) = a$. The integrating factor is:

$\mu(t) = e^{\int -4 dt} = e^{-4t}$

Step 3: Multiply both sides by the integrating factor

Multiply both sides of the differential equation by the integrating factor $e^{-4t}$:

$e^{-4t} \frac{dP}{dt} - 4e^{-4t} P = a e^{-4t}$

This simplifies to:

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

Step 4: Integrate both sides

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

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

On the left side, the derivative and integral cancel out, leaving:

$P e^{-4t} = \frac{a}{-4} e^{-4t} + C$

where $C$ is the constant of integration.

Step 5: Solve for $P$

Multiply both sides by $e^{4t}$ to solve for $P$:

$P = \frac{a}{-4} + C e^{4t}$

Simplifying:

$P = -\frac{a}{4} + C e^{4t}$

Final Solution:

The solution to the differential equation is:

$P = -\frac{a}{4} + C e^{4t}$

Would you like further details or have any other questions?

Here are 5 related questions to expand on this topic:

1. How do you find the integrating factor for a linear first-order differential equation in general?
2. What is the significance of the constant of integration in differential equations?
3. How would the solution change if $a = 0$?
4. How can you verify that the solution is correct by substituting it back into the original equation?
5. What other techniques can be used to solve first-order differential equations?

Tip: For linear differential equations, always check if the equation is in standard form before solving. This makes finding the integrating factor straightforward.