Find the general solution to the first order linear differential equation
dy/dt +4y = cos(2t)

Please use A to denote the constant of integration. Recall that e11t can be represented as exp(11*t) or as e^(11*t). Click the question mark in the answer panel for more information.


The first-order linear differential equation given is:

\[
\frac{dy}{dt} + 4y = \cos(2t)
\]

To solve this, we will use the method of integrating factors. The general form of a first-order linear differential equation is:

\[
\frac{dy}{dt} + p(t) y = g(t)
\]

where \( p(t) = 4 \) and \( g(t) = \cos(2t) \).

### Step 1: Find the integrating factor
The integrating factor \( \mu(t) \) is given by:

\[
\mu(t) = e^{\int p(t) \, dt} = e^{\int 4 \, dt} = e^{4t}
\]

### Step 2: Multiply the differential equation by the integrating factor
Multiplying both sides of the equation by \( e^{4t} \):

\[
e^{4t} \frac{dy}{dt} + 4 e^{4t} y = e^{4t} \cos(2t)
\]

The left-hand side of the equation is now the derivative of \( y e^{4t} \):

\[
\frac{d}{dt} \left( y e^{4t} \right) = e^{4t} \cos(2t)
\]

### Step 3: Integrate both sides
Integrate both sides with respect to \( t \):

\[
y e^{4t} = \int e^{4t} \cos(2t) \, dt
\]

We now solve the integral on the right-hand side. This can be done using integration by parts or the method of undetermined coefficients. The integral of \( e^{4t} \cos(2t) \) is:

\[
\int e^{4t} \cos(2t) \, dt = \frac{e^{4t}(4 \cos(2t) + 2 \sin(2t))}{20}
\]

Thus, we have:

\[
y e^{4t} = \frac{e^{4t}(4 \cos(2t) + 2 \sin(2t))}{20} + A
\]

### Step 4: Solve for \( y \)
Now, divide both sides by \( e^{4t} \):

\[
y = \frac{4 \cos(2t) + 2 \sin(2t)}{20} + A e^{-4t}
\]

Simplifying the coefficient:

\[
y = \frac{1}{5} \cos(2t) + \frac{1}{10} \sin(2t) + A e^{-4t}
\]

### General Solution:
The general solution to the differential equation is:

\[
y(t) = \frac{1}{5} \cos(2t) + \frac{1}{10} \sin(2t) + A e^{-4t}
\]

Would you like more details on any step or additional questions?

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Here are 5 related questions for you to consider:

1. How do you determine the integrating factor for other types of differential equations?
2. Can this method of integrating factors be applied to non-linear differential equations?
3. What is the role of the constant of integration in the general solution?
4. How would you solve a differential equation with variable coefficients?
5. How does the solution change if the forcing term \( g(t) \) is a different function, such as \( e^{2t} \)?

**Tip:** Always double-check your work when integrating, especially when using methods like integration by parts.

Find the general solution to the first order linear differential equation dy/dt + 4y = cos(2t).

Learn how to solve the first-order linear differential equation dy/dt + 4y = cos(2t) using the method of integrating factors. This step-by-step solution covers integration by parts and the use of trigonometric functions. Suitable for college-level students studying differential equations.

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