Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

Answers without the blur.

Just sign up for free and you're in.

Short Answer

Solve the given initial value problem. $y'' + 2 y' + 2 y = 0; y (0) = 2, y' (0) = 1$

The solution of the given initial value $y'' + 2 y' + 2 y = 0$ is $y (t) = e^{- t} (2 cost + 3 sint)$ when $y (0) = 2$ and $y' (0) = 1$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Initial value problem.

An initial value problem is an ordinary differential equation with a given initial condition. The solution of an ordinary differential equation is known as a general solution which consists of an arbitrary constant. The value of an arbitrary constant can be obtained by using the initial condition.

Step 2: Finding the general solution.

Given differential equation is $y'' + 2 y' + 2 y = 0 .$

Then the auxiliary equation is $r^{2} + 2 r + 2 = 0 .$

role="math" localid="1654075580000" $r = \frac{- 2 \pm \sqrt{2^{2} - 4 \times 1 \times 2}}{2 \times 1} r = \frac{- 2 \pm \sqrt{4 - 8}}{2} r = \frac{- 2 \pm \sqrt{- 4}}{2} r = \frac{- 2 \pm 2 i}{2} r = - 1 \pm i$

Therefore, the general solution is:

$\begin{matrix} y (t) = e^{- 1 \times t} (c_{1} \cos (t) + c_{2} \sin (t)) \\ y (t) = e^{- t} (c_{1} \cos (t) + c_{2} \sin (t)) \end{matrix}$

Step 3: Finding the values of c1 and c2

Given initial conditions are $y (0) = 2$ and $y' (0) = 1$

$\begin{matrix} y (0) = e^{- 0} (c_{1} \cos (0) + c_{2} \sin (0)) \\ c_{1} = 2 \end{matrix}$

And

$y' (t) = - e^{- t} (c_{1} cost + c_{2} sint) + e^{- t} (- c_{1} sint + c_{2} cost)$

Then we have:

$y' (0) = - e^{- 0} (c_{1} \cos 0 + c_{2} \sin 0) + e^{- 0} (- c_{1} \sin 0 + c_{2} \cos 0) - c_{1} + c_{2} = 1$

Substitute $c_{1}$ in the above equation

$\begin{array}{l} - 2 + c_{2} = 1 \\ c_{2} = 3 \end{array}$

Therefore, the solution is $y (t) = e^{- t} (2 cost + 3 sint)$ .

Find Study Materials for

Create Study Materials

Select your language

Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Solve the given initial value problem. $y'' + 2 y' + 2 y = 0; y (0) = 2, y' (0) = 1$

Step by Step Solution

Step 1: Initial value problem.

Step 2: Finding the general solution.

Step 3: Finding the values of c1 and c2

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Solve the given initial value problem. y''+2y'+2y=0;y(0)=2,y'(0)=1

Step by Step Solution

Step 1: Initial value problem.

Step 2: Finding the general solution.

Step 3: Finding the values of c1 and c2

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

Solve the given initial value problem. $y'' + 2 y' + 2 y = 0; y (0) = 2, y' (0) = 1$