6. Use the Laplace transform to solve the following IVP. The table is on the next page. y" + 3y' - 10y = 8(x) with y(0) = 0 and y'(0) = 0.
6. Use the Laplace transform to solve the following IVP. The table is on the next page. y" + 3y' - 10y = 8(x) with y(0) = 0 and y'(0) = 0.
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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![**Laplace Transform and Initial Value Problem (IVP) - Educational Resource**
---
**Title: Solving Initial Value Problems with Laplace Transform**
**Introduction:**
In this section, we will explore the application of the Laplace transform to solve initial value problems (IVPs). The use of Laplace transforms is a powerful technique for turning differential equations into algebraic equations, which are often easier to solve.
**Example Problem:**
Consider the second-order differential equation given by:
\[ y'' + 3y' - 10y = \delta(x) \quad \text{with} \quad y(0) = 0 \quad \text{and} \quad y'(0) = 0. \]
Here, \( \delta(x) \) represents the Dirac delta function, and the initial conditions are \( y(0) = 0 \) and \( y'(0) = 0 \).
**Steps to Solve Using Laplace Transform:**
1. **Apply the Laplace Transform:**
Convert the given differential equation into its Laplace transform form. Use the linearity property of the transform.
2. **Utilize Initial Conditions:**
Substitute the given initial conditions into the transformed equation to simplify the algebraic equation.
3. **Solve for Y(s):**
Solve the algebraic equation for the Laplace variable \( Y(s) \).
4. **Apply Inverse Laplace Transform:**
Once \( Y(s) \) is found, use the inverse Laplace transform to obtain \( y(t) \), the solution in the time domain.
**Reference Table:**
You will need a table of Laplace transforms and their inverses for standard functions to assist with steps in the solution process. This table, often provided in textbooks or on the next page of your academic resources, lists common transforms which can be directly used to find solutions quickly.
---
**Graphical Representation:**
A diagram or graph may provide a visual interpretation of the solution or the behavior of \( y(t) \) over time. Specifically, a graph showing \( y(t) \) versus \( t \) can illustrate the dynamic response of the solution due to the impulse function \( \delta(x) \).
---
By following this structured approach and leveraging the Laplace transform table, we can systematically solve complex differential equations in a streamlined fashion.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff0feac75-9cf9-420d-995e-ce89c47a4906%2F6b835c93-5809-410c-9ed2-3f27ce4445b8%2Fq93lr4v_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Laplace Transform and Initial Value Problem (IVP) - Educational Resource**
---
**Title: Solving Initial Value Problems with Laplace Transform**
**Introduction:**
In this section, we will explore the application of the Laplace transform to solve initial value problems (IVPs). The use of Laplace transforms is a powerful technique for turning differential equations into algebraic equations, which are often easier to solve.
**Example Problem:**
Consider the second-order differential equation given by:
\[ y'' + 3y' - 10y = \delta(x) \quad \text{with} \quad y(0) = 0 \quad \text{and} \quad y'(0) = 0. \]
Here, \( \delta(x) \) represents the Dirac delta function, and the initial conditions are \( y(0) = 0 \) and \( y'(0) = 0 \).
**Steps to Solve Using Laplace Transform:**
1. **Apply the Laplace Transform:**
Convert the given differential equation into its Laplace transform form. Use the linearity property of the transform.
2. **Utilize Initial Conditions:**
Substitute the given initial conditions into the transformed equation to simplify the algebraic equation.
3. **Solve for Y(s):**
Solve the algebraic equation for the Laplace variable \( Y(s) \).
4. **Apply Inverse Laplace Transform:**
Once \( Y(s) \) is found, use the inverse Laplace transform to obtain \( y(t) \), the solution in the time domain.
**Reference Table:**
You will need a table of Laplace transforms and their inverses for standard functions to assist with steps in the solution process. This table, often provided in textbooks or on the next page of your academic resources, lists common transforms which can be directly used to find solutions quickly.
---
**Graphical Representation:**
A diagram or graph may provide a visual interpretation of the solution or the behavior of \( y(t) \) over time. Specifically, a graph showing \( y(t) \) versus \( t \) can illustrate the dynamic response of the solution due to the impulse function \( \delta(x) \).
---
By following this structured approach and leveraging the Laplace transform table, we can systematically solve complex differential equations in a streamlined fashion.
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