Let's solve this step by step.

Part (a): Taking the Laplace transform of the differential equation

The given differential equation is: $4y'' + 5y' + 17y = 2\cos(2t),$ with initial conditions $y(0) = 0$ and $y'(0) = 0$.

Applying Laplace transform

Using the Laplace transform properties:

1. $\mathcal{L}\{y''(t)\} = s^2Y(s) - sy(0) - y'(0)$
2. $\mathcal{L}\{y'(t)\} = sY(s) - y(0)$
3. $\mathcal{L}\{y(t)\} = Y(s)$
4. $\mathcal{L}\{\cos(2t)\} = \frac{s}{s^2 + 4}$

Since $y(0) = 0$ and $y'(0) = 0$, the Laplace transform simplifies to: [ 4[s^2Y(s)] + 5[sY(s)] + 17Y(s) = \frac{2s}{s^2 + 4}. ]

Factor $Y(s)$ out: $Y(s) \big[ 4s^2 + 5s + 17 \big] = \frac{2s}{s^2 + 4}.$

Solve for $Y(s)$: $Y(s) = \frac{2s}{(4s^2 + 5s + 17)(s^2 + 4)}.$

This is the Laplace transform of $y(t)$.

Part (b): Express $y(t)$ using the convolution integral

We will write $y(t)$ in terms of a convolution integral without performing partial fraction decomposition.

The Laplace transform solution is: $Y(s) = \frac{2s}{(4s^2 + 5s + 17)(s^2 + 4)}.$

Using the convolution theorem: $Y(s) = \mathcal{L}\{f_1(t)\} \cdot \mathcal{L}\{f_2(t)\} \implies y(t) = \int_0^t f_1(v)f_2(t-v) \, dv,$ where:

1. $f_1(t)$ is the inverse Laplace transform of $\frac{1}{4s^2 + 5s + 17}$,
2. $f_2(t)$ is the inverse Laplace transform of $\frac{2s}{s^2 + 4}$.

Finding $f_2(t)$:

The inverse Laplace transform of $\frac{2s}{s^2 + 4}$ is: $f_2(t) = 2\cos(2t).$

Finding $f_1(t)$:

The term $\frac{1}{4s^2 + 5s + 17}$ requires finding the inverse Laplace transform. However, instead of explicitly solving, we represent $f_1(t)$ in terms of its transform.

Convolution integral:

$y(t) = \int_0^t f_1(v) \cdot 2\cos[2(t-v)] \, dv.$

This is the required expression for $y(t)$.

Summary

1. $\mathcal{L}\{y(t)\}(s) = \frac{2s}{(4s^2 + 5s + 17)(s^2 + 4)}$.
2. $y(t) = \int_0^t f_1(v) \cdot 2\cos[2(t-v)] \, dv$, where $f_1(t) = \mathcal{L}^{-1}\bigg(\frac{1}{4s^2 + 5s + 17}\bigg)$.

Let me know if you'd like me to expand or solve any part further!

Related Questions

1. How do you compute the inverse Laplace transform for $\frac{1}{4s^2 + 5s + 17}$?
2. What is the convolution theorem in the Laplace transform?
3. How does partial fraction decomposition help in solving Laplace-transformed equations?
4. How can this method be extended to non-homogeneous equations with different initial conditions?
5. What are the practical applications of solving differential equations using Laplace transforms?

Tip

When working with Laplace transforms, always double-check the initial conditions to simplify the algebraic manipulation of terms.