How can i solve this exercise? **Title:** Solving Initial Value Problems Using Laplace Transform**Objective:** Use the Laplace transform to solve the given initial value problem.**Problem Statement:**\[y'' - 4y' + 4y = t^3 e^{2t}, \quad y(0) = 0, \quad y'(0) = 0\]**Explanation:**In this problem, we are given a second-order linear differential equation with constant coefficients and a non-homogeneous term \( t^3 e^{2t} \). The initial conditions provided are \( y(0) = 0 \) and \( y'(0) = 0 \).To solve this initial value problem using the Laplace transform, we will follow these key steps:1. **Take the Laplace transform** of both sides of the differential equation. 2. **Apply the initial conditions** to simplify the resulting algebraic equation.3. **Solve for \( Y(s) \)**, the Laplace transform of the unknown function \( y(t) \).4. **Take the inverse Laplace transform** to obtain \( y(t) \).This approach leverages the properties of the Laplace transform to convert the differential equation into an algebraic equation that is easier to solve.**Step-by-Step Solution:**1. **Apply the Laplace transform** to both sides of the differential equation: \[ \mathcal{L}\{ y'' \} - 4\mathcal{L}\{ y' \} + 4\mathcal{L}\{ y \} = \mathcal{L}\{ t^3 e^{2t} \} \]2. Using the properties of Laplace transforms: \[ \mathcal{L}\{ y'' \} = s^2 Y(s) - s y(0) - y'(0) \] \[ \mathcal{L}\{ y' \} = s Y(s) - y(0) \] \[ \mathcal{L}\{ y \} = Y(s) \]3. Substituting the initial conditions \( y(0) = 0 \) and \( y'(0) = 0 \) into the transformed equation: \[ s^2 Y(s) - 4s Y

Name: How can i solve this exercise? **Title:** Solving Initial Value Proble
Uploaded: 2024-03-20T07:06:27.000Z
Channel: Bartleby
Description: How can i solve this exercise? **Title:** Solving Initial Value Problems Using Laplace Transform**Objective:** Use the Laplace transform to solve the given initial value problem.**Problem Statement:**\[y'' - 4y' + 4y = t^3 e^{2t}, \quad y(0) = 0, \qu