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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 4: Q24E (page 172)

Solve the given initial value problem. $y'' + 9 y = 0;$ $y (0) = 1,$ $y' (0) = 1$

Short Answer

Expert verified

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 $y' (0) = 1$ .

Step by step solution

01

Complex conjugate roots.

If the auxiliary equation has complex conjugate roots $伪卤颈尾$ , then the general solution is given as:

$y (t) = c_{1} e^{伪迟} 肠辞蝉尾迟 + c_{2} e^{伪迟} 蝉颈苍尾迟$

02

Finding the roots of the auxiliary equation.

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

Then the auxiliary equation is;

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

Therefore, the general solution is:

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

03

Finding the values of c1 and c2

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

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

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

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

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" $y (t) = \cos 3 t + \frac{1}{3} \sin 3 t$

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Most popular questions from this chapter

Find a particular solution to the differential equation. $y'' + 3 y = - 9$

Solve the given initial value problem. $w'' - 4 w' + 2 w = 0;$ $w (0) = 0,$ $w' (0) = 1$

A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation. $y'' + y' = 1, 鈥夆赌夆赌夆赌夆赌夆赌� y_{p} (t) = t$

A mass鈥搒pring system is driven by a sinusoidal external force $g (t) = 5 sint$ . The mass equals 1, the spring constant equals 3, and the damping coefficient equals 4. If the mass is initially located at $y (0) = \frac{1}{2}$ and at rest, i.e., $y' (0) = 0$ , find its equation of motion.

Vibrating Spring with Damping. Using the model for a vibrating spring with damping discussed in Example $3$

$(a)$ Find the equation of motion for the vibrating spring with damping if $m = 10 kg, b = 60 kg / \sec, k = 250 kg / \sec^{2}, y (0) = 0 . 3 m,$ and $y' (0) = - 0 . 1 m / \sec$ .

$(b)$ After how many seconds will the mass in part $(a)$ first cross the equilibrium point?

$(c)$ Find the frequency of oscillation for the spring system of part $(a)$ .

$(d)$ Compare the results of problems $32$ and $33$ determine what effect the damping has on the frequency of oscillation. What other effects does it have on the solution?

See all solutions

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