Q19P By using Laplace transforms, sol... [FREE SOLUTION]

91影视

Chapter 8: Q19P (page 443)

By using Laplace transforms, solve the following differential equations subject to the given initial conditions. $y^{'}' + y^{'} - 5 y = e^{2} t, 鈥� y_{0} = 1, y_{0}^{'} = 2$

Short Answer

Expert verified

The given differential equation's solution is $y = e^{2 t}$ .

Step by step solution

Given information from question

The differential equation is $y^{'}' + y^{'} - 5 y = e^{2} t, 鈥� y_{0} = 1, y_{0}^{'} = 2 .$ .

Laplace transform

Application of Laplace transform:

It's used to simplify complicated differential equations by using polynomials. It's used to convert derivatives into numerous domain variables, then utilise the Inverse Laplace transform to convert the polynomials back to the differential equation.

Take the Laplace of given equation

The given equation is

$y^{'}' + y^{'} - 5 y = e^{2 t}$

Take the Laplace of equation

$L \{y^{'}' + y^{'} - 5 y\} = L \{e^{2 t}\} L \{y^{'}'\} + L \{y'\} - 5 L \{y\} = L \{e^{2 t}\} p^{2} L \{y\} - p y_{o} - y'_{0} + [p L \{y\} - y_{0}] - 5 L \{y\} = \frac{1}{p - 2} (p^{2} + p - 5) L \{y\} - (p + 1) y_{o} - y'_{0} = \frac{1}{p - 2}$

Further solve

$(p^{2} + p - 5) L \{y\} - (p + 1) (1) - (2) = \frac{1}{p - 2} (p^{2} + p - 5) L \{y\} = \frac{1}{p - 2} + p + 2 L \{y\} = \frac{1}{p - 2 (p^{2} + p - 5)} + \frac{2}{(p^{2} + p - 5)} + \frac{p}{(p^{2} + p - 5)} L \{y\} = (\frac{1}{p - 2})$

The inverse Laplace is

$y = L^{- 1} \{\frac{1}{(p - 2)}\} y = e^{2 t}$

Thus, the given differential equation's solution is $y = e^{2 t}$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

By using Laplace transforms, solve the following differential equations subject to the given initial conditions. $y^{'}' + y^{'} - 5 y = e^{2} t, 鈥� y_{0} = 1, y_{0}^{'} = 2$

Short Answer

Step by step solution

Given information from question

Laplace transform

Take the Laplace of given equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Turning Points in Physics

Particle Model of Matter

Quantum Physics

Work Energy and Power

Modern Physics

Scientific Method Physics

Study anywhere. Anytime. Across all devices.

91影视

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.y''+y'-5y=e2t,鈥�y0=1,y0'=2

Short Answer

Step by step solution

Given information from question

Laplace transform

Take the Laplace of given equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Turning Points in Physics

Particle Model of Matter

Quantum Physics

Work Energy and Power

Modern Physics

Scientific Method Physics

Study anywhere. Anytime. Across all devices.

By using Laplace transforms, solve the following differential equations subject to the given initial conditions. $y^{'}' + y^{'} - 5 y = e^{2} t, 鈥� y_{0} = 1, y_{0}^{'} = 2$