https://file.aiquickdraw.com/mathbot/file/17265044151246fq54wo0.jpg

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.

Solve the differential equation dP/dt = 4P + a, where a is a non-zero constant. Use C for any constant of integration.

Step-by-step solution to the differential equation dP/dt = 4P + a, where a is a constant. This example demonstrates the integration technique for solving linear first-order differential equations, including the use of the integrating factor. Suitable for college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation