The given differential equation is:

$y'' - 2y = e^{-t}$ with initial conditions:

$y(0) = 0, \quad y'(0) = 0$

We need to solve this differential equation using the Laplace transform and find the inverse Laplace transform.

Step 1: Take the Laplace transform of the equation

First, recall the Laplace transform of derivatives:

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

Apply the Laplace transform to both sides of the equation:

$\mathcal{L}(y'') - 2\mathcal{L}(y) = \mathcal{L}(e^{-t})$

Using the initial conditions $y(0) = 0$ and $y'(0) = 0$, the Laplace transform of the equation becomes:

$s^2 Y(s) - 2 Y(s) = \frac{1}{s+1}$

Step 2: Solve for $Y(s)$

Factor out $Y(s)$ on the left-hand side:

$(s^2 - 2) Y(s) = \frac{1}{s+1}$

Solve for $Y(s)$:

$Y(s) = \frac{1}{(s^2 - 2)(s+1)}$

Step 3: Perform partial fraction decomposition

We need to decompose $Y(s) = \frac{1}{(s^2 - 2)(s+1)}$ into simpler terms that are easier to inverse transform. To do this, let's rewrite $s^2 - 2$ as $(s - \sqrt{2})(s + \sqrt{2})$, and then apply partial fraction decomposition.

Let me compute the decomposition and find the inverse Laplace transform.The inverse Laplace transform of the function $Y(s) = \frac{1}{(s^2 - 2)(s+1)}$ is:

$y(t) = \frac{1}{4} \left[ (\sqrt{2} + 2)e^t - 4e^{\sqrt{2} t} + (2 - \sqrt{2}) e^{(1 + 2\sqrt{2})t} \right] e^{-(1 + \sqrt{2})t} \cdot \text{Heaviside}(t)$

This is the solution $y(t)$ of the given differential equation, which is valid for $t \geq 0$ due to the presence of the Heaviside function $\text{Heaviside}(t)$.

Would you like further details on how the partial fraction decomposition and inverse Laplace transform were applied?

Related Questions:

1. What is the general process for solving differential equations using Laplace transforms?
2. How does the Heaviside function impact solutions in Laplace transforms?
3. Can you explain the role of initial conditions in solving differential equations using the Laplace method?
4. What are the properties of Laplace transforms for exponential functions?
5. How is partial fraction decomposition used in solving Laplace transform problems?

Tip:

When dealing with second-order differential equations, the Laplace transform is particularly useful as it converts differentiation into algebraic operations, simplifying the process of solving.