Fundamentals of Differential Equations

R. Kent Nagle

Chapter 4

Linear Second-Order Equations - all with Video Answers

Educators

Section 1

Introduction: The Mass-Spring Oscillator

Problem 1

Verify that for $$b=0$$ and $$F_{\operatorname{ext}}(t)=0$$, equation (3) has a
solution of the form

$$y(t)=\cos \omega t, \text { where } \omega=\sqrt{k / m}$$

Sajin Shajee

Sajin Shajee

Numerade Educator

Problem 2

If $$F_{\mathrm{ext}}(t)=0$$, equation (3) becomes

$$m y^{\prime \prime}+b y^{\prime}+k y=0$$

For this equation, verify the following:

(a) If $$y(t)$$ is a solution, so is $$c y(t)$$, for any constant $$C$$.
(b) If $$y_{1}(t)$$ and $$y_{2}(t)$$ are solutions,so is their sum $$y_{1}(t)+y_{2}(t)$$.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 3

Show that if $$F_{\mathrm{ext}}(t)=0, m=1, k=9$$, and $$b=6$$, then equation (3) has the 'critically damped' solutions $$y_{1}(t)=e^{-3 t}$$ and $$y_{2}(t)=t e^{-3 t}$$. What is the limit of these solutions as $$t \rightarrow \infty$$?

Sajin Shajee

Numerade Educator

Problem 4

Verify that $$y=\sin 3 t+2 \cos 3 t$$ is a solution to the initial value problem

$$2 y^{\prime \prime}+18 y=0 ; \quad y(0)=2, \quad y^{\prime}(0)=3$$

Find the maximum of $$|y(t)| \text { for }-\infty<t<\infty$$.

Carson Merrill

Numerade Educator

Problem 5

Verify that the exponentially damped sinusoid $$y(t)=e^{-3 t} \sin (\sqrt{3} t)$$ is a solution to equation (3) if $$F_{\mathrm{exx}}(t)=0, m=1, b=6$$, and $$k=12$$. What is the limit of this solution as $$t \rightarrow \infty ?$$

Sajin Shajee

Numerade Educator

Problem 6

An external force $$F(t)=2 \cos 2 t$$ is applied to a mass spring system with $$m=1, b=0$$ and $$k=4$$, which is initially at rest; i.e., $$y(0)=0, y^{\prime}(0)=0$$. Verify that $$y(t)=\frac{1}{2} t \sin 2 t$$ gives the motion of this spring.What will eventually (as t increases) happen to the spring?

A M

Numerade Educator

Problem 7

$$y^{\prime \prime}+2 y^{\prime}+4 y=5 \sin 3 t, \quad \Omega=3$$

A M

Numerade Educator

Problem 8

$$y^{\prime \prime}+2 y^{\prime}+5 y=-50 \sin 5 t, \quad \Omega=5$$

A M

Numerade Educator

Problem 9

$$y^{\prime \prime}+2 y^{\prime}+4 y=6 \cos 2 t+8 \sin 2 t, \quad \Omega=2$$

Sajin Shajee

Numerade Educator

Problem 10

Undamped oscillators that are driven at resonance have unusual (and nonphysical) solutions.
(a) To investigate this, find the synchronous solution $A \cos \Omega t+B \sin \Omega t$ to the generic forced oscillator
equation
(7)
$$
m y^{\prime \prime}+b y^{\prime}+k y=\cos \Omega t
$$
(b) Sketch graphs of the coefficients $A$ and $B,$ as functions of $\Omega,$ for $m=1, b=0,1,$ and $k=25$
(c) Now set $b=0$ in your formulas for $A$ and $B$ and resketch the graphs in part (b), with $m=1,$ and $k=25 .$ What happens at $\Omega=5 ?$ Notice that the amplitudes of the synchronous solutions grow without bound as $\Omega$ approaches 5 .
(d) Show directly, by substituting the form $A \cos \Omega t+$ $B \sin \Omega t$ into equation $(7),$ that when $b=0$ there are no synchronous solutions if $\Omega=\sqrt{k / m}$
(e) Verify that $(2 m \Omega)^{-1} t \sin \Omega t$ solves equation (7) when $b=0$ and $\Omega=\sqrt{k / m}$. Notice that this nonsynchronous solution grows in time, without bound.
Clearly one cannot neglect damping in analyzing an oscillator forced at resonance, because otherwise the solutions, as shown in part (e), are nonphysical. This behavior will be studied later in this chapter.

Sajin Shajee

Numerade Educator