Chapter 1: Q19E (page 1)

In Problem 19, solve the given initial value problem
$\begin{array}{l} y''' - y'' - 4 y' + 4 y = 0 \\ y (0) = - 4 \\ y' (0) = - 1 \\ y'' (0) = - 19 \end{array}$

Short Answer

Expert verified

The solution is C $y (t) = e^{t} - 3 e^{2 t} - 2 e^{- 2 t}$

Step by step solution

Basic differentiation

The Sum rule says the derivative of a sum of functions is the sum of their derivatives. The Difference rule says the derivative of a difference of functions is the difference of their derivatives.

Solving by basic differentiation:

We will do the following question on the basis of basic differentiation ;

$\begin{array}{l} r^{3} - r^{2} - 4 r + 4 = 0 \\ r^{3} - 4 r^{2} + 7 r - 6 = (r - 1) (r - 2) (r + 2) = 0 \\ y (t) = c_{1} e^{t} + c_{2} e^{2 t} + c_{3} e^{- 2 t} \\ y' (t) = c_{1} e^{t} + 2 c_{2} e^{2 t} - 2 c_{3} e^{- 2 t} \\ y'' (t) = c_{1} e^{t} + 4 c_{2} e^{2 t} + 4 c_{3} e^{- 2 t} \\ y (0) = - 4 \\ y' (0) = - 1 \\ y'' (0) = - 19 \\ y (t) = e^{t} - 3 e^{2 t} - 2 e^{- 2 t} \end{array}$

Hence, the final answer is:

$y (t) = e^{t} - 3 e^{2 t} - 2 e^{- 2 t}$

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In Problem 19, solve the given initial value problem
$\begin{array}{l} y''' - y'' - 4 y' + 4 y = 0 \\ y (0) = - 4 \\ y' (0) = - 1 \\ y'' (0) = - 19 \end{array}$

Short Answer

Step by step solution

Basic differentiation

Solving by basic differentiation:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Applied Mathematics

Logic and Functions

Discrete Mathematics

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

In Problem 19, solve the given initial value problemy'''−y''−4y'+4y=0y(0)=−4y'(0)=−1y''(0)=−19

Short Answer

Step by step solution

Basic differentiation

Solving by basic differentiation:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Applied Mathematics

Logic and Functions

Discrete Mathematics

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

In Problem 19, solve the given initial value problem
$\begin{array}{l} y''' - y'' - 4 y' + 4 y = 0 \\ y (0) = - 4 \\ y' (0) = - 1 \\ y'' (0) = - 19 \end{array}$