The Chemistry Maths Book

Erich Steiner

Chapter 12

Second-order differential equations. Constant coefficients - all with Video Answers

Educators

Chapter Questions

Problem 1

. Show that $e^{-2 x}$ and $e^{2 \pi / 3}$ are particular solutions of the differential equation $3 y^{\prime \prime}+4 y^{\prime}-4 y=0$.

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Problem 2

Show that $e^{3 x}$ and $x e^{3 x}$ are particular solutions of the differential equation $y^{n}-6 y^{\prime}+9 y=0$.

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Problem 3

Show that $\cos 2 x$ and $\sin 2 x$ are particular solutions of the differential equation $y^{\prime \prime}+4 y=0$. Write down the general solution of the differential equation in
4. Exercise $1 .$
5. Exercise 2 .
6. Exercise 3 .

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Problem 4

Exercise $1 .$

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Problem 5

Exercise 2 .

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Problem 6

Exercise 3 .

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Problem 7

$y^{\prime \prime}-y^{\prime}-6 y=0$

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Problem 8

$2 y^{\prime \prime}-8 y^{\prime}+3 y=0$

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Problem 9

$y^{\prime \prime}-8 y^{\prime}+16 y=0$

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Problem 10

$4 y^{\prime \prime}+12 y+9 y=0$

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Problem 11

. $y^{\prime \prime}+4 y^{\prime}+5 y=0$

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Problem 12

$y^{\prime \prime}+3 y^{\prime}+5 y=0$

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Problem 13

$\frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}-2 x=0 ; \quad x(0)=1, \quad \frac{d x}{d t}(0)=0$

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Problem 14

$\frac{d^{2} x}{d t^{2}}+6 \frac{d x}{d t}+9 x=0 ; \quad x(1)=0, \quad \frac{d x}{d t}(1)=1$

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Problem 15

$\frac{d^{2} x}{d t^{2}}+9 x=0 ; \quad x(\pi / 3)=0, \frac{d x}{d t}(\pi / 3)=-1$

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Problem 16

$\frac{d^{2} x}{d t^{2}}-2 \frac{d x}{d t}+2 x=0 ; \quad x(0)=1, \frac{d x}{d t}(0)=0$

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Problem 17

Solve the boundary value problems:
17. $\frac{d^{2} y}{d x^{2}}+4 \frac{d y}{d x}+8 y=0 ; \quad y(\pi / 2)=-1, y(3 \pi / 4)=1$

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Problem 18

$\frac{d^{2} y}{d x^{2}}+9 y=0 ; \quad y=0$ when $x=0, y=1$ when $x=\pi / 2$

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Problem 19

$\frac{d^{2} y}{d x^{2}}+8 \frac{d y}{d x}+16 y=0 ; \quad y(0)=0, y(1)=1$

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Problem 20

$\frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}-2 y=0 ; \quad y(0)=2, y \rightarrow 0$ as $x \rightarrow \infty$

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Problem 21

Solve $\frac{d^{2} \theta}{d t^{2}}+a^{2} \theta=0$ subject to the condition $\theta(t+2 \pi \tau)=\theta(t)$.

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Problem 22

Given that the general solution of the equation of motion $m \bar{x}=-k x$ for the harmonic oscillator is $x(t)=a \cos \omega t+b \sin \omega t$, where $\omega=\sqrt{k / m}$, (i) show that the solution can be written in the form $x(t)=A \cos (\omega t-\delta)$, where $A$ is the amplitude of the vibration and $\delta$ is the phase angle, and express $A$ and $\delta$ in terms of $a$ and $b$; (ii) find the amplitude and phase angle for the initial conditions $x(0)=1, \dot{x}(0)=\omega$,

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Problem 23

Solve the equation of motion for the harmonic oscillator with initial conditions $x(0)=0$, $\dot{x}(0)=u_{0}$

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Problem 24

For the particle in a box, find the nodes and sketch the graph of the wave function $\psi_{n}$ for (i) $n=4$ and (ii) $n=5$.

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Problem 25

(i) Solve the Schrödinger equation (12.44) for the particle in a box of length $l$ with potential-energy function $V=0$ for $-l / 2 \leq x \leq+l / 2, V=\infty$ for $x \leq-l / 2$ and $x \geq+l / 2$.
(ii) Show that the solutions $\psi_{n}$ are even functions of $x$ when $n$ is odd and odd functions when $n$ is even. (iii) Show that the solutions are the same as those given by (12.53) if $x$ is replaced by $x+l / 2$, except for a possible change of sign.

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Problem 26

For the particle in the box in Section 12.6, show that wave functions $\psi_{n}(x)=\sqrt{\frac{2}{l}} \sin \frac{n \pi x}{l}$ for $n=1$ and $n=2$ are (i) normalized, (ii) orthogonal.

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Problem 27

For the particle in a ring show that wave functions $\psi_{n}(\theta)=1 / \sqrt{2 \pi} e^{i n \theta}$ for $n=3$ and $n=4$ are (i) normalized, (ii) orthogonal.

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Problem 28

The diagrams of Figure $12.8$ are maps of the signs and nodes of some real wave functions (12.71) for the particle in a ring. Draw the corresponding diagrams for (i) $n=\pm 3$,
(ii) $n=\pm 4$.

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Problem 29

Verify that equation (12.72) and its solutions (12.74) are transformed into (12.62) and (12.65) by means of the change of variable $\theta=x / r$.

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Problem 30

Find a particular solution of the differential equation $y^{\prime \prime}-y^{\prime}-6 y=2+3 x$.

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Problem 31

$y^{\prime \prime}-y^{\prime}-6 y=2+3 x$

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Problem 32

$y^{n}-8 y^{\prime}+16 y=1-4 x^{3}$

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Problem 33

$y^{\prime \prime}-y^{\prime}-6 y=2 e^{-3 x}$

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Problem 34

$y^{N}-y^{\prime}-2 y=3 e^{-x}$

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Problem 35

$y^{n \prime}-8 y^{\prime}+16 y=e^{4 x}$

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Problem 36

$y^{\prime \prime}-y^{\prime}-6 y=2 \cos 3 x$

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Problem 37

$y^{n}+4 y=3 \sin 2 x$

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Problem 38

$y^{N}-y^{\prime}-6 y=2+3 x+2 e^{-3 x}+2 \cos 3 x$

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Problem 39

An RLC-circuit contains a resistor (resistance $R$ ), an inductor (inductance $L$ ), and a capacitor (capacitance $C$ ) connected in series with a source of e.m.f. $E$. (i) Use Kirchhoff's voltage law (Section $11.7$ ) to show that the current $I(t)$ in the circuit is given by the inhomogeneous equation
$$
L \frac{d^{2} I}{d t^{2}}+R \frac{d I}{d t}+\frac{I}{C}=\frac{d E}{d t}
$$
(ii) Find the solution of the homogeneous equation (for $d E / d t=0$ ), and confirm that it decays exponentially as $t \rightarrow \infty$.
(iii) Show that the particular integral for the periodic e.m.f. $E(t)=E_{0} \sin \omega t$ is $I_{p}(t)=I_{0} \sin (\omega t-\delta)$
where $I_{0}=\frac{E_{0}}{\sqrt{R^{2}+S^{2}}}, \tan \delta=\frac{S}{R}$, and $S=\omega L-\frac{1}{\omega C}$.

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