Let's apply the initial condition \( y(0) = 9 \) to the general solution we found:

### General Solution
The general solution is:

\[
y(t) = C e^{-2t} + 11\sin(t) + 2\cos(t)
\]

### Step 1: Substitute the initial condition
We substitute \( t = 0 \) and \( y(0) = 9 \) into the general solution:

\[
y(0) = C e^{-2(0)} + 11\sin(0) + 2\cos(0)
\]

Simplifying this:

\[
9 = C \cdot 1 + 11 \cdot 0 + 2 \cdot 1
\]
\[
9 = C + 2
\]

### Step 2: Solve for \( C \)
Solve the equation for \( C \):

\[
C = 9 - 2
\]
\[
C = 7
\]

### Final Solution
Substitute \( C = 7 \) back into the general solution:

\[
y(t) = 7e^{-2t} + 11\sin(t) + 2\cos(t)
\]

This is the solution to the initial value problem.

---

Would you like more details on any specific part? Here are 5 related questions:

1. How would the solution change if the initial condition was \( y(1) = 5 \) instead of \( y(0) = 9 \)?
2. How can you verify that this solution satisfies the original differential equation?
3. What are some methods for solving differential equations if undetermined coefficients don't work?
4. How does the initial condition affect the particular solution?
5. Can this method be applied to systems of differential equations?

**Tip:** Always check your final solution by substituting it back into the original differential equation to ensure it satisfies both the equation and the initial condition.

Learn how to apply the initial condition y(0) = 9 to a differential equation solution step-by-step. Understand the use of exponential and trigonometric functions in solving initial value problems. Suitable for advanced high school 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