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Connect and share knowledge within a single location that is structured and easy to search. ## Initial value problem using Laplace Transform

I have to solve the IVP: $$\begin{cases} y''+by'-cy=1\\ y(0)=y_0\\ y'(0)=y'_0 \end{cases}$$

Suppose $$Y(s)=\frac{s^2+2s+1}{s^3+3s^2+2s}$$

We have to find $b,c,y_0$ and $y'_0$

I have found that: $$(s^2+s+1)Y(s)-(s+1)y_0-y'_0=1/s$$

How do I proceed further According to the given ODE $y''+by'-cy=1$ , your last line should be $$(s^2+bs-c)Y(s)-(s+b)y_0-y'_0=1/s$$ which implies $$Y(s)=\frac{y_0s^2+(by_0+y'_0)s+1}{s^3+bs^2-cs}.$$ Now, by comparing it with $Y(s)=\frac{s^2+2s+1}{s^3+3s^2+2s}$ , we are able to find $b$ , $c$ , $y_0$ and $y'_0$ easily: $$b=3,c=-2, y_0=1, y'_0=-1.$$

You have that the Laplace transform of the solution $y(t)$ is $Y(s)$ . So the inverse Laplace transform of $Y(s)$ gives us the solution $y(t) =\frac{e^{-2t}+1}{2}$ . From this we can calculate $y(0)=1$ and $y'(0) =-1$ . Now sustitute the expression of $y(t)$ in the equation and you'll get $$e^{-2t}(4-2b+c) -(2+c) =0.$$ Since $e^{-2t}$ and $1$ are linearly independent we Have that $b=3$ and $c=-2$ . Not the answer you're looking for browse other questions tagged laplace-transform initial-value-problems inverse-laplace ..

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Laplace transform

## Solve the initial value problem using Laplace transforms. y"+y=f(t) , y(0)=0 , y'(0)=1 Lennie Carroll

Solve the initial value problem using Laplace transforms. y " + y = f ( t ) , y ( 0 ) = 0 , y ′ ( 0 ) = 1 Here f ( t ) = { 0 0 ≤ t < 3 π 1 t ≥ 3 π Arnold Odonnell

Do you have a similar question? ## New Questions in Differential Equations

The Laplace transform of the product of two functions is the product of the Laplace transforms of each given function. True or False

The Laplace transform of u ( t − 2 ) is (a) 1 s + 2 (b) 1 s − 2 (c) e 2 s s ( d ) e − 2 s s ??

1 degree on celsius scale is equal to A) 9 5 degree on fahrenheit scale B) 5 9 degree on fahrenheit scale C) 1 degree on fahrenheit scale D) 5 degree on fahrenheit scale

The Laplace transform of t e t is A. s ( s + 1 ) 2 B. 1 ( s − 1 ) 2 C. s ( s + 1 ) 2 D. s ( s − 1 )

What is the Laplace transform of t cos ⁡ t into the s domain?

Find the general solution of the given differential equation: y ″ − 2 y ′ + y = 0

The rate at which a body cools is proportional to the difference in  temperature between the body and its surroundings. If a body in air  at 0℃ will cool from 200℃ 𝑡𝑜 100℃ in 40 minutes, how many more  minutes will it take the body to cool from 100℃ 𝑡𝑜 50℃ ?

A body falls from rest against a resistance proportional to the velocity at any instant. If the limiting velocity is 60fps and the body attains half that velocity in 1 second, find the initial velocity.

What's the correct way to go about computing the Inverse Laplace transform of this? − 2 s + 1 ( s 2 + 2 s + 5 ) I Completed the square on the bottom but what do you do now? − 2 s + 1 ( s + 1 ) 2 + 4

How to find inverse Laplace transform of the following function? X ( s ) = s s 4 + 1 I tried to use the definition: f ( t ) = L − 1 { F ( s ) } = 1 2 π i lim T → ∞ ∫ γ − i T γ + i T e s t F ( s ) d s or the partial fraction expansion but I have not achieved results.

How do i find the lapalace transorm of this intergral using the convolution theorem? ∫ 0 t e − x cos ⁡ x d x

How can I solve this differential equation? : x y   d x - ( x 2 + 1 )   d y = 0

Find the inverse Laplace transform of s 2 − 4 s − 4 s 4 + 8 s 2 + 16

inverse laplace transform - with symbolic variables: F ( s ) = 2 s 2 + ( a − 6 b ) s + a 2 − 4 a b ( s 2 − a 2 ) ( s − 2 b ) My steps: F ( s ) = 2 s 2 + ( a − 6 b ) s + a 2 − 4 a b ( s + a ) ( s − a ) ( s − 2 b ) = A s + a + B s − a + C s − 2 b + K K = 0 A = F ( s ) ∗ ( s + a )

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## Browse Course Material

Course info, instructors.

• Prof. Arthur Mattuck
• Prof. Haynes Miller
• Dr. Jeremy Orloff
• Dr. John Lewis

• Mathematics

## As Taught In

• Mathematics Differential Equations Linear Algebra

## Learning Resource Types

Differential equations, laplace transform: solving initial value problems.

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## Session Overview

Session activities.

• Laplace Transform: Solving Initial Value Problems: Introduction (PDF)
• Laplace Transform Table (PDF)
• Table Entries: Derivative Rules (PDF)
• Precise Definition of Laplace Inverse (PDF)
• Laplace: Solving Initial Value Problems (PDF)
• IVP’s and t-translation (PDF)
• IVP’s: Longer Examples (PDF)

Watch the problem solving video:

• Laplace: Solving ODE’s

Complete the practice problems:

• Practice Problems 29 (PDF)
• Practice Problems 29 Solutions (PDF)

## Check Yourself

Complete the problem sets:

Problem Set Part I Problems (PDF)

Problem Set Part I Solutions (PDF)

Problem Set Part II Problems (PDF)

Problem Set Part II Solutions (PDF)  #### VIDEO

1. Inverse Laplace Transform

2. Laplace Transforms using Derived Formulas Part 1

3. Application of Laplace Transforms to Differential Equations-1

4. Euler's Method-Numerical technique to solve Initial Value Problem!

5. laplace problems _part 3

6. Euler's Modified Method-Numerical technique to solve Initial Value Problem!

1. Initial value problem using Laplace Transform

We have to find b,c,y₀ and y'₀. I have found that:(s^2+s+1)*Y(s)-(s+1)y₀-y'₀=1/s. How do I proceed further

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Now that we know how to find a Laplace transform, it is time to use it to solve differential equations. The key feature of the Laplace transform that makes it a tool for solving differential equations is that

3. Use the Laplace transform to solve the initial value problem

Find step-by-step Engineering solutions and the answer to the textbook question Use the Laplace transform to solve the initial value problem. y' – 2y = 1 – t; y(0) = 4

4. Laplace Transform: Solving Initial Value Problems

This section provides materials for a session on operations on the simple relation between the Laplace transform of a function and the Laplace transform of its derivative

5. Solve the initial value problem using Laplace transforms. y+y=f(t)

To solve the initial value problem using Laplace transforms, we first take the Laplace transform of both sides of the given equations. Using the property of the Laplace transform for derivatives, we have

6. Laplace Transform: Solving Initial Value Problems

This section provides materials for a session on operations on the simple relation between the Laplace transform of a function and the Laplace transform of its derivative