Sure, I can help you solve these differential equations using Laplace transforms. Let's break down each problem one by one.

Problem 4(a)

The differential equation is: $y'' + 3y' - y = x$ with initial conditions $y(0) = 1$ and $y'(0) = -1$.

1. Apply the Laplace transform to both sides of the equation: $\mathcal{L}\{y'' + 3y' - y\} = \mathcal{L}\{x\}$

2. Use the properties of Laplace transforms: $\mathcal{L}\{y''\} = s^2Y(s) - sy(0) - y'(0)$ $\mathcal{L}\{3y'\} = 3(sY(s) - y(0))$ $\mathcal{L}\{y\} = Y(s)$

3. Substitute the initial conditions and the transforms into the equation: $s^2Y(s) - s \cdot 1 - (-1) + 3[sY(s) - 1] - Y(s) = \mathcal{L}\{x\}$

4. Simplify the equation: $(s^2 + 3s - 1)Y(s) - s + 1 - 3 = \frac{1}{s^2}$ $(s^2 + 3s - 1)Y(s) = \frac{1}{s^2} + s - 2$ $Y(s) = \frac{1}{s^2} \cdot \frac{1}{s^2 + 3s - 1}$

5. Solve for $Y(s)$: Decompose $\frac{1}{s^2 + 3s - 1}$ into partial fractions and then use the inverse Laplace transform.

Problem 4(b)

The differential equation is: $y'' + 2y' + 5y = 3e^{-x} \sin(x)$ with initial conditions $y(0) = 0$ and $y''(0) = 3$.

1. Apply the Laplace transform to both sides of the equation: $\mathcal{L}\{y'' + 2y' + 5y\} = \mathcal{L}\{3e^{-x} \sin(x)\}$

2. Use the properties of Laplace transforms: $\mathcal{L}\{y''\} = s^2Y(s) - sy(0) - y'(0)$ $\mathcal{L}\{2y'\} = 2(sY(s) - y(0))$ $\mathcal{L}\{5y\} = 5Y(s)$ $\mathcal{L}\{3e^{-x} \sin(x)\} = \frac{3}{(s+1)^2 + 1}$

3. Substitute the initial conditions and the transforms into the equation: $s^2Y(s) - sy(0) - y'(0) + 2(sY(s) - y(0)) + 5Y(s) = \frac{3}{(s+1)^2 + 1}$

4. Simplify the equation: $(s^2 + 2s + 5)Y(s) = \frac{3}{(s+1)^2 + 1}$ $Y(s) = \frac{3}{((s+1)^2 + 1)(s^2 + 2s + 5)}$

5. Solve for $Y(s)$: Decompose $\frac{3}{((s+1)^2 + 1)(s^2 + 2s + 5)}$ into partial fractions and then use the inverse Laplace transform.

For detailed steps and solving, the partial fraction decomposition and inverse Laplace transform steps are needed.

Do you need detailed steps for the inverse Laplace transforms and partial fraction decomposition?

Suggested Questions

1. How do you perform partial fraction decomposition?
2. What are the properties of the Laplace transform?
3. How do you find the inverse Laplace transform?
4. How to solve a system of linear differential equations using Laplace transforms?
5. How to handle non-homogeneous differential equations with Laplace transforms?

Tip

Remember that the initial conditions are crucial in using Laplace transforms for solving differential equations, as they simplify the transformed equation and make it easier to solve for $Y(s)$.