https://file.aiquickdraw.com/mathbot/file/1726022871875cbjm97i1.jpg

The differential equation presented is:

\[
y' - 5y = 3 \sin(2t)
\]

This is a first-order linear differential equation, and we will solve it using the method of integrating factor.

### Step 1: Standard form
The standard form for a linear differential equation is:

\[
y' + P(t)y = Q(t)
\]

We rewrite the given equation as:

\[
y' - 5y = 3 \sin(2t)
\]

In standard form:

\[
y' + (-5)y = 3 \sin(2t)
\]

Here, \( P(t) = -5 \) and \( Q(t) = 3 \sin(2t) \).

### Step 2: Finding the integrating factor
The integrating factor \( \mu(t) \) is given by:

\[
\mu(t) = e^{\int P(t) \, dt} = e^{\int -5 \, dt} = e^{-5t}
\]

### Step 3: Multiply the equation by the integrating factor
Now we multiply the whole equation by \( e^{-5t} \):

\[
e^{-5t} y' - 5 e^{-5t} y = 3 \sin(2t) e^{-5t}
\]

The left-hand side becomes the derivative of \( e^{-5t} y \):

\[
\frac{d}{dt} \left( e^{-5t} y \right) = 3 \sin(2t) e^{-5t}
\]

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

\[
e^{-5t} y = \int 3 \sin(2t) e^{-5t} \, dt
\]

We need to compute this integral on the right-hand side.

### Step 5: Solving the integral
This is an integral of the form \( \int e^{at} \sin(bt) \, dt \), which can be solved using integration by parts or a known formula. The solution for \( \int e^{at} \sin(bt) \, dt \) is:

\[
\int e^{at} \sin(bt) \, dt = \frac{e^{at}}{a^2 + b^2} \left( a \sin(bt) - b \cos(bt) \right)
\]

Substitute \( a = -5 \) and \( b = 2 \):

\[
\int e^{-5t} \sin(2t) \, dt = \frac{e^{-5t}}{(-5)^2 + 2^2} \left( -5 \sin(2t) - 2 \cos(2t) \right)
\]

\[
= \frac{e^{-5t}}{25 + 4} \left( -5 \sin(2t) - 2 \cos(2t) \right)
= \frac{e^{-5t}}{29} \left( -5 \sin(2t) - 2 \cos(2t) \right)
\]

Multiply this result by 3 to account for the original equation:

\[
3 \int e^{-5t} \sin(2t) \, dt = \frac{3e^{-5t}}{29} \left( -5 \sin(2t) - 2 \cos(2t) \right)
\]

### Step 6: General solution
Thus,

\[
e^{-5t} y = \frac{3e^{-5t}}{29} \left( -5 \sin(2t) - 2 \cos(2t) \right) + C
\]

Finally, multiply both sides by \( e^{5t} \) to solve for \( y \):

\[
y = \frac{3}{29} \left( -5 \sin(2t) - 2 \cos(2t) \right) + C e^{5t}
\]

### Final Answer:
\[
y = \frac{-15 \sin(2t) - 6 \cos(2t)}{29} + C e^{5t}
\]

Let me know if you'd like further clarifications!

---

Here are five related questions to extend the topic:
1. How do you find the integrating factor for a linear differential equation?
2. What is the general form of a first-order linear differential equation?
3. How do you solve a differential equation using the method of undetermined coefficients?
4. What are the conditions for the uniqueness of the solution to a first-order differential equation?
5. How does variation of parameters work in solving non-homogeneous differential equations?

**Tip**: Always simplify your differential equation first to match a known method, like integrating factor or undetermined coefficients, which can significantly speed up solving time.

Find the general solution of the differential equation: y' - 5y = 3 sin(2t).

Learn how to find the general solution to the first-order linear differential equation y' - 5y = 3sin(2t). This involves using the method of integrating factors and solving trigonometric integrals. Suitable for university-level students studying differential equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation