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. $y'' + 9 y = 0;$ $y (0) = 1,$ $y' (0) = 1$

The solution of the given initial value $y'' + 9 y = 0$ is $y (t) = \cos 3 t + \frac{}{} \sin 3 t$ when $y (0) = 1$ and $y' (0) = 1$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Complex conjugate roots.

If the auxiliary equation has complex conjugate roots $α \pm iβ$ , then the general solution is given as:

$y (t) = c_{1} e^{αt} cosβt + c_{2} e^{αt} sinβt$

Step 2: Finding the roots of the auxiliary equation.

Given differential equation is $y'' + 9 y = 0$

Then the auxiliary equation is;

$\begin{matrix} r^{2} + 9 = 0 \\ r^{2} = - 9 \\ r = \pm \sqrt{- 9} \\ r = \pm 3 i \end{matrix}$

Therefore, the general solution is:

$\begin{array}{l} y (t) = e^{0 \times t} (c_{1} \cos (3 t) + c_{2} \sin (3 t)) \\ = c_{1} \cos (3 t) + c_{2} \sin (3 t)) \end{array}$

Step 3: Finding the values of c1 and c2

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

$\begin{array}{l} y (0) = (c_{1} \cos (3 \times 0) + c_{2} \sin (3 \times 0)) \\ c_{1} = 1 \end{array}$

And $y' (t) = (- c_{1} \sin (3 t) + c_{2} \cos (3 t))$

Then $y' (0) = (- 3 c_{1} \sin (3 \times 0) + 3 c_{2} \cos (3 \times 0))$

role="math" localid="1654843939522" $\begin{matrix} 3 c_{2} = 1 \\ c_{2} = \frac{1}{3} \end{matrix}$

Therefore, the solution is role="math" localid="1654843966518" $y (t) = \cos 3 t + \frac{1}{3} \sin 3 t$

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. $y'' + 9 y = 0;$ $y (0) = 1,$ $y' (0) = 1$

Step by Step Solution

Step 1: Complex conjugate roots.

Step 2: Finding the roots of the auxiliary equation.

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''+9y=0;y(0)=1,y'(0)=1

Step by Step Solution

Step 1: Complex conjugate roots.

Step 2: Finding the roots of the auxiliary equation.

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. $y'' + 9 y = 0;$ $y (0) = 1,$ $y' (0) = 1$