**Understanding Laplace Transforms in Differential Equations**In the study of differential equations, the Laplace transform is often employed to transform differential equations into algebraic equations, which are simpler to solve. Here we consider the Laplace transforms of derivatives of a function \( x(t) \).### Laplace Transform of Derivatives**Given:**1. **Second Derivative:** \[ \mathcal{L}\{x''\} = s^2 \cdot x(s) - x(0) \cdot s - x'(0) \] This formula represents the Laplace transform of the second derivative of \( x(t) \). Here, \( x'' \) denotes the second derivative of \( x \) with respect to \( t \). \( s \) is a complex frequency parameter, \( x(s) \) is the Laplace transform of the function \( x(t) \), \( x(0) \) is the initial value of \( x \) at \( t = 0 \), and \( x'(0) \) is the initial value of the first derivative of \( x \).2. **First Derivative:** \[ \mathcal{L}\{x'\} = s \cdot x(s) - x(0) \] This represents the Laplace transform of the first derivative of \( x(t) \). Similar to the second derivative formula, \( s \) is the complex frequency parameter, \( x(s) \) is the Laplace transform of \( x(t) \), and \( x(0) \) is the initial value of \( x \).3. **Original Function:** \[ \mathcal{L}\{x\} = x(s) \] This simple expression states that the Laplace transform of the function \( x(t) \) is \( x(s) \).By utilizing these transformations, solving differential equations can be converted into solving algebraic equations, which are generally easier to manage. The initial conditions \( x(0) \) and \( x'(0) \) play a crucial role in these transformed equations.For more detailed explanations and examples, visit our Differential Equations section on our educational website. **Title: Solving Initial Value Problems (IVP) using Laplace Transforms****Objective:**Learn how to solve initial value problems involving differential equations using Laplace Transforms.**Problem Statement:****Solve the given IVP using Laplace Transforms.***Given Differential Equation:*\[ x'' + 9x = \cos(3t) \]*Initial Conditions:*\[ x(0) = 0, \quad x'(0) = 2 \]**Task:***Solve for \( x(s) \) and STOP! Express \( x(s) \) as a single fraction.***Step-by-Step Solution:**1. **Apply the Laplace Transform to both sides of the differential equation.** - Recall that: \[ \mathcal{L} \{ x''(t) \} = s^2 X(s) - sx(0) - x'(0) \] \[ \mathcal{L} \{ x'(t) \} = s X(s) - x(0) \] \[ \mathcal{L} \{ x(t) \} = X(s) \] \[ \mathcal{L} \{ \cos(3t) \} = \frac{s}{s^2 + 9} \]2. **Substitute the Laplace Transforms into the differential equation:** Given: \[ x'' + 9x = \cos(3t) \] After applying Laplace Transforms, we get: \[ s^2 X(s) - sx(0) - x'(0) + 9X(s) = \frac{s}{s^2 + 9} \]3. **Substitute the initial conditions \( x(0) = 0 \) and \( x'(0) = 2 \):** \[ s^2 X(s) - 0 - 2 + 9X(s) = \frac{s}{s^2 + 9} \] Simplifies to: \[ (s^2 + 9)X(s) - 2 = \frac{s}{s^2 + 9} \]4. **Solve for \( X(s) \):** Move \(-2\) to the other side: \[ (s

Given IVP is x'' + 9x = cos3t, x0=0, x'0=2

Answered: Solve the given IVP using Laplace…

Solve the given IVP using Laplace Transforms. Solve for x(s) and STOP! (Express x(s) as a single fraction.) x" + 9x cos(3t), x(0)=0, x'(0) = 2

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

See similar textbooks

Related questions

Q: 3y" + 6y + 3y = 9, y(0) = 0, /(0) = 6 %3D

Q: x+ 5x' + 6x=8 (1-4), x(0) = 20, x'(0) = 7 Solve the above IVP using Laplace Transforms and then find…

A: we have to find the solution of the given differential equation using Laplace transformation.

Q: showo haw [co(t)+ 3sine) ]= vioet coslt- antan( 3)

Q: (i) Beginning with the homogeneous version of ODE, find the characteristic equation by letting…

Q: Let (t) = Find '(t) '(t)=<

Q: Use the method of Laplace transforms to solve the given initial value problem. Here, x' and y'…

Q: Compute r"(t) and r'''(t) for the following function. r(t) = (7t2 + 4,t + 3,6)

Q: Solve the following ODE 3t. y" – 6y' + 9y = t²et; y(0) = 0, y'(0) = %3D

A: Given The given ODE is y"-6y'+9y=t2e3t. y0=0, y'0=1 Apply formula…

Q: A dilation T sends A = (2,3) to A’ = (5,4), and it sends B = (3,−1) to B’ = (7,−4). Where does it…

A: We have, A dilation T sends A = (2,3) to A’ = (5,4) and it sends B = (3,−1) to B’ = (7,−4) Where…

Q: Suppose r' (t) = (cos(t), et, 2t) and r(0) = (1, 0, 3). Find r(t).

Q: y = tan o (sin 67E-TT) find dy dt ()

Q: Solve the IVP x”+9x = U(t-2)sin[3(t-2)], x(0)=2, x’(0)=0 using Laplace Transforms. Identify xc , xp…

A: The solution is given as

Q: Let r(t) = Find 7¹ (t) r¹' (t) = < " V

Q: Determine r (x) if r(x) = (arccot(x))".

Q: Use Laplace transforms to solve the following initial value problem. x"' + 25x = sin 8t, x(0)=0,…

Q: f"(t)-4f(t)=sin2t where f(0)=1 and f'(0)=-2 Use the Laplace transform to solve the equation

Question

**Understanding Laplace Transforms in Differential Equations**

In the study of differential equations, the Laplace transform is often employed to transform differential equations into algebraic equations, which are simpler to solve. Here we consider the Laplace transforms of derivatives of a function \( x(t) \).

### Laplace Transform of Derivatives

**Given:**

1. **Second Derivative:**
\[
\mathcal{L}\{x''\} = s^2 \cdot x(s) - x(0) \cdot s - x'(0)
\]
This formula represents the Laplace transform of the second derivative of \( x(t) \). Here, \( x'' \) denotes the second derivative of \( x \) with respect to \( t \). \( s \) is a complex frequency parameter, \( x(s) \) is the Laplace transform of the function \( x(t) \), \( x(0) \) is the initial value of \( x \) at \( t = 0 \), and \( x'(0) \) is the initial value of the first derivative of \( x \).

2. **First Derivative:**
\[
\mathcal{L}\{x'\} = s \cdot x(s) - x(0)
\]
This represents the Laplace transform of the first derivative of \( x(t) \). Similar to the second derivative formula, \( s \) is the complex frequency parameter, \( x(s) \) is the Laplace transform of \( x(t) \), and \( x(0) \) is the initial value of \( x \).

3. **Original Function:**
\[
\mathcal{L}\{x\} = x(s)
\]
This simple expression states that the Laplace transform of the function \( x(t) \) is \( x(s) \).

By utilizing these transformations, solving differential equations can be converted into solving algebraic equations, which are generally easier to manage. The initial conditions \( x(0) \) and \( x'(0) \) play a crucial role in these transformed equations.

For more detailed explanations and examples, visit our Differential Equations section on our educational website.

**Title: Solving Initial Value Problems (IVP) using Laplace Transforms**

**Objective:**
Learn how to solve initial value problems involving differential equations using Laplace Transforms.

**Problem Statement:**

**Solve the given IVP using Laplace Transforms.**

*Given Differential Equation:*
\[ x'' + 9x = \cos(3t) \]

*Initial Conditions:*
\[ x(0) = 0, \quad x'(0) = 2 \]

**Task:**

*Solve for \( x(s) \) and STOP! Express \( x(s) \) as a single fraction.*

**Step-by-Step Solution:**

1. **Apply the Laplace Transform to both sides of the differential equation.**

- Recall that:
\[ \mathcal{L} \{ x''(t) \} = s^2 X(s) - sx(0) - x'(0) \]
\[ \mathcal{L} \{ x'(t) \} = s X(s) - x(0) \]
\[ \mathcal{L} \{ x(t) \} = X(s) \]
\[ \mathcal{L} \{ \cos(3t) \} = \frac{s}{s^2 + 9} \]

2. **Substitute the Laplace Transforms into the differential equation:**

Given:
\[ x'' + 9x = \cos(3t) \]

After applying Laplace Transforms, we get:
\[ s^2 X(s) - sx(0) - x'(0) + 9X(s) = \frac{s}{s^2 + 9} \]

3. **Substitute the initial conditions \( x(0) = 0 \) and \( x'(0) = 2 \):**

\[ s^2 X(s) - 0 - 2 + 9X(s) = \frac{s}{s^2 + 9} \]
Simplifies to:
\[ (s^2 + 9)X(s) - 2 = \frac{s}{s^2 + 9} \]

4. **Solve for \( X(s) \):**

Move \(-2\) to the other side:
\[ (s

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

Similar questions

If p(z) = p'(z) = LV² T Cos w⁹ w2 dw then
Compute cos(4x) -∞ (x²+2x+2)' dx.
Use the formula {tf(t))} (s) = (-1)^({f}(s)) to help determine the following the expressions. dn ds" (a) {t cos bt} (b) {t² cos bt} Click here to view the table of Laplace transforms. (a) At cos bt}(s) =

Recommended textbooks for you

Calculus: Early Transcendentals

Calculus

ISBN:

9781285741550

Author:

James Stewart

Publisher:

Cengage Learning