**Title: Solving Initial Value Problems Using Laplace Transforms****Problem Statement:**1. Use the method of Laplace Transforms to solve the following initial value problem: $ y'' + 3y' = 7, $ with the initial conditions: $ y(0) = 1 $ and $ y'(0) = -3 $.**Instructions:**- Apply the Laplace Transform to both sides of the differential equation.- Utilize the initial conditions to formulate the transformed equation.- Solve the algebraic equation obtained in the Laplace domain.- Apply the inverse Laplace Transform to find the solution $ y(t) $ in the time domain.**Explanation:**The Laplace Transform is a very effective mathematical tool for solving linear ordinary differential equations, particularly with given initial conditions. This method converts the differential equations into algebraic equations, which are often simpler to handle. The given problem is a second-order linear differential equation with constant coefficients and specific initial conditions.**Steps:**1. **Transform the Equation:** - Take the Laplace Transform of both sides of the given differential equation $y'' + 3y' = 7$. 2. **Include Initial Conditions in the Transform:** - Apply the initial conditions $y(0) = 1$ and $y'(0) = -3$ while solving the transformed equation. 3. **Solve Algebraic Equation:** - Solve the resulting algebraic equation in the Laplace domain to find the Laplace Transform of $ y(t) $. 4. **Inverse Laplace Transform:** - Finally, apply the inverse Laplace Transform to the solution obtained in the Laplace domain to get $ y(t) $ in the time domain.---For a detailed step-by-step solution and technical explanations, follow through with the educational modules on solving differential equations using Laplace Transforms. This problem is aimed to enhance your understanding of the practical applications of Laplace Transforms in solving initial value problems in systems described by differential equations.

For the solution of a given initial value problem with boundary conditions given, take Laplace…

Answered: 1. Use the method of Laplace Transforms…

1. Use the method of Laplace Transforms to solve the following initial value problem. y" + 3y = 7, y(0) = 1, y/ (0) = –3

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Title: Solving Initial Value Problems Using Laplace Transforms**

**Problem Statement:**

1. Use the method of Laplace Transforms to solve the following initial value problem:

$ y'' + 3y' = 7, $

with the initial conditions:

$ y(0) = 1 $ and $ y'(0) = -3 $.

**Instructions:**

- Apply the Laplace Transform to both sides of the differential equation.
- Utilize the initial conditions to formulate the transformed equation.
- Solve the algebraic equation obtained in the Laplace domain.
- Apply the inverse Laplace Transform to find the solution $ y(t) $ in the time domain.

**Explanation:**

The Laplace Transform is a very effective mathematical tool for solving linear ordinary differential equations, particularly with given initial conditions. This method converts the differential equations into algebraic equations, which are often simpler to handle. The given problem is a second-order linear differential equation with constant coefficients and specific initial conditions.

**Steps:**

1. **Transform the Equation:**
- Take the Laplace Transform of both sides of the given differential equation $y'' + 3y' = 7$.

2. **Include Initial Conditions in the Transform:**
- Apply the initial conditions $y(0) = 1$ and $y'(0) = -3$ while solving the transformed equation.

3. **Solve Algebraic Equation:**
- Solve the resulting algebraic equation in the Laplace domain to find the Laplace Transform of $ y(t) $.

4. **Inverse Laplace Transform:**
- Finally, apply the inverse Laplace Transform to the solution obtained in the Laplace domain to get $ y(t) $ in the time domain.

---

For a detailed step-by-step solution and technical explanations, follow through with the educational modules on solving differential equations using Laplace Transforms. This problem is aimed to enhance your understanding of the practical applications of Laplace Transforms in solving initial value problems in systems described by differential equations.

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