https://file.aiquickdraw.com/mathbot/file/1727668345017ombnhcgv.jpg

The given second-order linear differential equation is:

\[
y'' - 4y' + 29y = 0
\]

We will solve this using the characteristic equation method.

### Step 1: Write the characteristic equation
The characteristic equation corresponding to the differential equation \(y'' - 4y' + 29y = 0\) is obtained by assuming a solution of the form \(y = e^{rt}\), which leads to the equation:

\[
r^2 - 4r + 29 = 0
\]

### Step 2: Solve the characteristic equation
Solve the quadratic equation:

\[
r^2 - 4r + 29 = 0
\]

Using the quadratic formula:

\[
r = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(29)}}{2(1)}
\]

\[
r = \frac{4 \pm \sqrt{16 - 116}}{2}
\]

\[
r = \frac{4 \pm \sqrt{-100}}{2}
\]

\[
r = \frac{4 \pm 10i}{2}
\]

\[
r = 2 \pm 5i
\]

### Step 3: General solution
Since the roots are complex, the general solution of the differential equation is:

\[
y(t) = e^{2t}(c_1 \cos(5t) + c_2 \sin(5t))
\]

### Step 4: Match the solution with the options
The correct form from the options provided is:

\[
y(t) = e^{2t}(c_1 \cos(5t) + c_2 \sin(5t))
\]

Thus, the correct choice is:

\[
\boxed{y(t) = c_1 e^{2t} \cos(5t) + c_2 e^{2t} \sin(5t)}
\]

Would you like more details or have any questions? 

Here are some related questions to deepen your understanding:
1. What is the general form of the solution when the characteristic equation has complex roots?
2. How would the solution change if the roots of the characteristic equation were real?
3. Why do we express the complex exponential in terms of sine and cosine?
4. What is the significance of the imaginary part of the roots in oscillatory behavior?
5. How can initial conditions be used to determine the constants \(c_1\) and \(c_2\)?

**Tip:** For solving second-order differential equations, the nature of the characteristic equation's roots (real, repeated, or complex) dictates the form of the solution.

Find the general solution to the differential equation y'' - 4y' + 29y = 0

Learn how to find the general solution to the second-order differential equation y'' - 4y' + 29y = 0 using the characteristic equation method. This problem involves solving for complex roots and applying the general formula for solutions with real and imaginary parts. Suitable for undergraduate math 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