Analytical and Computational Methods of Advanced Engineering Mathematics

Grant B. Gustafson, Calvin H. Wilcox

Chapter 3

Ordinary Differential Equations of Higher Order - all with Video Answers

Educators

Section 1

Examples from Engineering and Physics

Problem 1

LRC Circuits. Determine the differential equation and initial conditions for the given circuit.
$R=100, L=1, E=5$ and no capacitor. Initial charge and current are zero.

Sajin Shajee

Sajin Shajee

Numerade Educator

Problem 2

LRC Circuits. Determine the differential equation and initial conditions for the given circuit.
$C=1 / 10, L=1, E=5$ and no resistor. Initial charge and current are zero.

Sajin Shajee

Numerade Educator

Problem 3

LRC Circuits. Determine the differential equation and initial conditions for the given circuit.
$R=100, C=1 / 10, L=1, E=5$. Initial charge and current are zero.

Sajin Shajee

Numerade Educator

Problem 4

LRC Circuits. Determine the differential equation and initial conditions for the given circuit.
$R=100, L=1, E=5$ and no capacitor. Initial charge is zero and the initial current is 5 .

Sajin Shajee

Numerade Educator

Problem 5

LRC Circuits. Determine the differential equation and initial conditions for the given circuit.
$C=1 / 10, L=1, E=5$ and no resistor. Initially the capacitor is charged to 0.05 farads and the current is initially zero.

Sajin Shajee

Numerade Educator

Problem 6

LRC Circuits. Determine the differential equation and initial conditions for the given circuit.
$R=100, C=1 / 10, L=1, E=5$. The capacitor is charged to 0.001 farads and an ammeter reads 3 amperes at time $t=0$.

Sajin Shajee

Numerade Educator

Problem 7

Mechanical Oscillators. Determine the differential equation and initial conditions for the given mechanical system. All units are cgs.
No damping is present, the mass is 10 and the Hooke's spring constant is 2 . No external force. At time $t=0$ the position and velocity are zero and one, respectively.

Carson Merrill

Numerade Educator

Problem 8

Mechanical Oscillators. Determine the differential equation and initial conditions for the given mechanical system. All units are cgs.
No damping is present, the mass is 10 and the Hooke's spring constant is 2 . The external force is 100 sin $\omega t$. At time $t=0$ the position and velocity are both zero.

Carson Merrill

Numerade Educator

Problem 9

Torsional Pendulum. Determine the differential equation and initial conditions for the given system. All units are cgs.
The moment of inertia is 1 , no damping is present and the elastic restoring force constant is $10^{-2}$. No external torque is present. The twist angle and velocity are initially one.

Joanna Josey

Numerade Educator

Problem 10

Torsional Pendulum. Determine the differential equation and initial conditions for the given system. All units are cgs.
The moment of inertia is 5 , the damping constant is 2 and the elastic restoring force constant is 1 . No external torque is present. The twist angle and velocity are initially one.

Joanna Josey

Numerade Educator

Problem 11

Torsional Pendulum. Determine the differential equation and initial conditions for the given system. All units are cgs.
The moment of inertia is 1 , the damping constant is 3 and the elastic restoring force constant is 2 . The external torque is $10 \sin \omega t$. The twist angle and velocity are initially zero.

Joanna Josey

Numerade Educator

Problem 12

Torsional Pendulum. Determine the differential equation and initial conditions for the given system. All units are cgs.
The moment of inertia is 1 , no damping is present and the elastic restoring force constant is 2 . No external torque is present. The twist angle is initially 90 degrees and the velocity is initially zero.

Joanna Josey

Numerade Educator

Problem 13

A string of length 10 creates a pendulum with attached mass 0.01 , all in $f p s$ units. The small oscillations of the pendulum begin by pulling the pendulum to angular position 7 degrees and releasing the mass.

Narayan Hari

Numerade Educator

Problem 14

A string of length 5 creates a pendulum with attached mass 10 , all in cgs units. The small oscillations of the pendulum begin by pulling the pendulum to angular position $\pi / 12$ radians and releasing the mass with velocity $1 \mathrm{~cm} / \mathrm{sec}$.

Urvashi Arora

Numerade Educator

Problem 15

The partial differential equation is

$$
\frac{\partial u}{\partial t}=\frac{\partial}{\partial x}\left(\sqrt{x} \frac{\partial u}{\partial x}\right)-x^2 \sin (x) u
$$

Sajin Shajee

Numerade Educator

Problem 16

The partial differential equation is

$$
\frac{\partial u}{\partial t}=\frac{\partial}{\partial x}\left(x^2 \frac{\partial u}{\partial x}\right)-\left(1+\sin ^2(x)\right) u+(1+x)^{3 / 2}
$$

Sajin Shajee

Numerade Educator

Problem 17

(Bessel Equation) $\ln r^2 R^{\prime \prime}(r)+r R^{\prime}(r)+\left(\omega^2 r^2-n^2\right) R(r)=0$, make the change of variables $x=\omega r$, $y(x)=R(r)$ to obtain the Bessel differential equation $x^2 y^{\prime \prime}+x y^{\prime}+\left(x^2-n^2\right) y=0$.

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Problem 18

(Steady-State Temperature in a Disk) Derive the differential equation $r^2 R^{\prime \prime}(r)+r R^{\prime}(r)-n^2 R(r)=0$ by substitution of the product solution $u=R(r) \Theta(\theta)$ into the partial differential equation $r^2 u_{r r}+r u_r+$ $u_{9 \theta}=0$. Assume that the separation constant $\lambda$ has the form $\lambda=n^2(\sec$ Chapter 8$)$.

Sajin Shajee

Numerade Educator

Problem 19

(Legendre Differential Equation) Explain why $\left(\left(1-x^2\right) y^{\prime}\right)^{\prime}+\mu y=0$ is equivalent to the equation $\left(1-x^2\right) y^{\prime \prime}-2 x y^{\prime}+\mu y=0$.

Raj Bala

Numerade Educator

Problem 20

(Euler's Differential Equation) Define variables $r=e^x, y(x)=R(r)$. Find relations for $r^2 d^2 R / d r^2$ and $r d R / d r$ in terms of $y$ and $x$. Use them to show that the Euler equation $r^2 \frac{d^2 R}{d r^2}+r \frac{d R}{d r}-\mu R=0$ is equivalent to the constant coefficient equation $y^{\prime \prime}-\mu y=0$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 21

(Multiloop Circuit) Verify directly that $q=A_1 e^{-t}+A_2 e^{-2 t}+A_3 e^{-3 t}$ is a solution of the loop equation $q^{\prime \prime \prime}+6 q^{\prime \prime}+11 q^{\prime}+6 q=0$.

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Problem 22

(Mechanical Oscillator) Verify directly that $x=A_1 e^{-t}+A_2 t e^{-t}+A_3 \cos t+A_4 \sin t$ is a solution of the oscillator equation $x^{\prime \prime \prime \prime}+2 x^{\prime \prime \prime}+2 x^{\prime \prime}+2 x^{\prime}+x=0$.

Nick Johnson

Numerade Educator

Problem 23

(Parallel LRC Circuit) Consider the parallel LRC circuit of Figure 11 with current source $i(t)$. Let $i_1$, $i_2, i_3$ and $i$ be the currents in the four branches, as shown, and let $u(t)=L i_3(t)$. Use Kirchhoffs laws to show that

$$
C u^{\prime \prime}+\frac{1}{R} u^{\prime}+\frac{1}{L} u=i(t)
$$

Lainey Roebuck

Numerade Educator

Problem 24

(Mass-String System) For the mass-string system of Figure 5, show that for small displacements $x$ the vertical force on the mass $m$ is $-2 T \sin \theta \approx-2 T \tan \theta$ and $\tan \theta=2 x / \ell$. Then use Newton's law to verify the differential equation (14).

Suzanne W.

Numerade Educator

Problem 25

(Two-Loop Circuit) Consider the two-loop electrical circuit of Figure 12 . Let $i_1, i_2$ be the electrical currents in the two branches, as shown, and let $q_1, q_2$ be the corresponding electrical charges on the plates of the two capacitors.

Use Kirchhoffs laws to derive a system of two differential equations for $q_1, q_2$. Then show that the system is isomorphic to the system (57), (58) for the mechanical oscillator of Figure 10.

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