To solve this differential equation using the Laplace transform, let us break it down step-by-step:

Problem Statement

We are solving $y'' + 9y = 0$, with initial conditions:

• $y(0) = y_0$,
• $y'(0) = y_1$.

Step 1: Apply the Laplace transform

Take the Laplace transform of both sides of the equation. Recall the Laplace transform rules:

1. $\mathcal{L}(y'(t)) = sY(s) - y(0)$,
2. $\mathcal{L}(y''(t)) = s^2Y(s) - sy(0) - y'(0)$.

Applying this, we get: $\mathcal{L}(y'') + 9\mathcal{L}(y) = 0.$

Substituting the Laplace transforms: $s^2Y(s) - sy(0) - y'(0) + 9Y(s) = 0.$

Simplify using the initial conditions $y(0) = y_0$ and $y'(0) = y_1$: $s^2Y(s) - sy_0 - y_1 + 9Y(s) = 0.$

Step 2: Solve for $Y(s)$

Group terms involving $Y(s)$: $Y(s)(s^2 + 9) = sy_0 + y_1.$

Solve for $Y(s)$: $Y(s) = \frac{sy_0 + y_1}{s^2 + 9}.$

Step 3: Take the inverse Laplace transform

To find $y(t)$, apply the inverse Laplace transform to $Y(s)$. First, rewrite $Y(s)$ as: $Y(s) = \frac{sy_0}{s^2 + 9} + \frac{y_1}{s^2 + 9}.$

Using the standard Laplace transform pairs:

1. $\mathcal{L}^{-1}\left(\frac{s}{s^2 + \omega^2}\right) = \cos(\omega t)$,
2. $\mathcal{L}^{-1}\left(\frac{1}{s^2 + \omega^2}\right) = \frac{\sin(\omega t)}{\omega}$,

we compute: $\mathcal{L}^{-1}\left(\frac{sy_0}{s^2 + 9}\right) = y_0 \cos(3t),$ $\mathcal{L}^{-1}\left(\frac{y_1}{s^2 + 9}\right) = \frac{y_1}{3} \sin(3t).$

Thus, the solution is: $y(t) = y_0 \cos(3t) + \frac{y_1}{3} \sin(3t).$

Step 4: Verify the answer

From the solution, we see that the correct answer is D: $y(t) = y_0 \cos(3t) + \frac{y_1}{3} \sin(3t).$

Would you like additional clarification or detailed explanations of any step?

Related Questions:

1. How do you handle initial conditions in Laplace transforms for higher-order equations?
2. What are the key differences between solving differential equations with Laplace transforms and other methods?
3. Why is the term $\frac{y_1}{3} \sin(3t)$ divided by 3 in this solution?
4. Can Laplace transforms be used for non-homogeneous differential equations?
5. What other real-world problems can be solved using Laplace transforms?

Tip:

Always check the initial conditions after solving the differential equation to ensure the solution satisfies them!