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Mathematical Methods for Physics and Engineering: A Comprehensive Guide

K. F. Riley, M. P. Hobson, S. J. Bence

Chapter 14

First-order ordinary differential equations - all with Video Answers

Educators

Chapter Questions

01:19

Problem 1

A radioactive isotope decays in such a way that the number of atoms present at a given time, $N(t)$, obeys the equation
$$
\frac{d N}{d t}=-\lambda N
$$
If there are initially $N_{0}$ atoms present, find $N(t)$ at later times.

Keshav Singh
Keshav Singh
Numerade Educator
03:18

Problem 2

Solve the following equations by separation of the variables:
(a) $y^{\prime}-x y^{3}=0$
(b) $y^{\prime} \tan ^{-1} x-y\left(1+x^{2}\right)^{-1}=0$;
(c) $x^{2} y^{\prime}+x y^{2}=4 y^{2}$.

Keshav Singh
Keshav Singh
Numerade Educator
15:32

Problem 3

Show that the following equations are either exact or can be made exact, and solve them:
(a) $y\left(2 x^{2} y^{2}+1\right) y^{\prime}+x\left(y^{4}+1\right)=0$;
(b) $2 x y^{\prime}+3 x+y=0$;
(c) $\left(\cos ^{2} x+y \sin 2 x\right) y^{\prime}+y^{2}=0$.

Gaurav Kalra
Gaurav Kalra
Numerade Educator
03:17

Problem 4

Find the values of $\alpha$ and $\beta$ that make
$$
F(x, y)=\left(\frac{1}{x^{2}+2}+\frac{\alpha}{y}\right) d x+\left(x y^{\beta}+1\right) d y
$$
an exact differential. For these values solve $F(x, y)=0$.

Sam Stansfield
Sam Stansfield
Numerade Educator
04:30

Problem 5

By finding a suitable integrating factor, solve the following equations:
(a) $\left(1-x^{2}\right) y^{\prime}+2 x y=\left(1-x^{2}\right)^{3 / 2}$;
(b) $y^{\prime}-y \cot x+\operatorname{cosec} x=0$;
(c) $\left(x+y^{3}\right) y^{\prime}=y$ (treat $y$ as the independent variable).

Aman Gupta
Aman Gupta
Numerade Educator
03:26

Problem 6

By finding an appropriate integrating factor, solve
$$
\frac{d y}{d x}=-\frac{2 x^{2}+y^{2}+x}{x y}.
$$

Keshav Singh
Keshav Singh
Numerade Educator
02:56

Problem 7

Find, in the form of an integral, the solution of the equation
$$
\alpha \frac{d y}{d t}+y=f(t)
$$
for a general function $f(t) .$ Find the specific solutions for
(a) $f(t)=H(t)$
(b) $f(t)=\delta(t)$
(c) $f(t)=\beta^{-1} e^{-t / \beta} H(t)$ with $\beta<\alpha$
For case (c), what happens if $\beta \rightarrow 0 ?$

Vikash Ranjan
Vikash Ranjan
Numerade Educator
05:57

Problem 8

An electric circuit contains a resistance $R$ and a capacitor $C$ in series, and a battery supplying a time-varying electromotive force $V(t) .$ The charge $q$ on the capacitor therefore obeys the equation
$$
R \frac{d q}{d t}+\frac{q}{C}=V(t).
$$
Assuming that initially there is no charge on the capacitor, and given that $V(t)=V_{0} \sin \omega t$, find the charge on the capacitor as a function of time.

Keshav Singh
Keshav Singh
Numerade Educator
02:21

Problem 9

Using tangential-polar coordinates (see exercise $2.20$ ), consider a particle of mass $m$ moving under the influence of a force $f$ directed towards the origin $\mathrm{O}$. By resolving forces along the instantaneous tangent and normal and making use of the result of exercise $2.20$ for the instantaneous radius of curvature, prove that
$$
f=-m v \frac{d v}{d r} \quad \text { and } \quad m v^{2}=f p \frac{d r}{d p}
$$
Show further that $h=m p v$ is a constant of the motion and that the law of force can be deduced from
$$
f=\frac{h^{2}}{p^{3}} \frac{d p}{d r}.
$$

Keshav Singh
Keshav Singh
Numerade Educator
02:21

Problem 10

Use the result of the previous exercise to find the law of force, acting towards the origin, under which a particle must move so as to describe the following trajectories:
(a) A circle of radius $a$ which passes through the origin;
(b) An equiangular spiral, which is defined by the property that the angle $\alpha$ between the tangent and the radius vector is constant along the curve.

Keshav Singh
Keshav Singh
Numerade Educator
05:58

Problem 11

Solve
$$
(y-x) \frac{d y}{d x}+2 x+3 y=0.
$$

Brittany Knowlton
Brittany Knowlton
Numerade Educator
04:23

Problem 12

A mass $m$ is accelerated by a time-varying force $\exp (-\beta t) v^{3}$, where $v$ is its velocity. It also experiences a resistive force $\eta v$, where $\eta$ is a constant, owing to its motion through the air. The equation of motion of the mass is therefore
$$
m \frac{d v}{d t}=\exp (-\beta t) v^{3}-\eta v.
$$
Find an expression for the velocity $v$ of the mass as a function of time, given that it has an initial velocity $v_{0}$.

Keshav Singh
Keshav Singh
Numerade Educator
05:43

Problem 13

Using the results about Laplace transforms given in chapter 13 for $d f / d t$ and $t f(t)$, show, for a function $y(t)$ that satisfies
$$
t \frac{d y}{d t}+(t-1) y=0
$$
with $y(0)$ finite, that $\bar{y}(s)=C(1+s)^{-2}$ for some constant $C$.
Given that
$$
y(t)=t+\sum_{n=2}^{\infty} a_{n} t^{n}
$$
determine $C$ and show that $a_{n}=(-1)^{n} / n !$. Compare this result with that obtained by integrating $\left(^{*}\right)$ directly.

Aidan Mcnabb
Aidan Mcnabb
Numerade Educator
01:48

Problem 14

Solve
$$
\frac{d y}{d x}=\frac{1}{x+2 y+1}.
$$

Keshav Singh
Keshav Singh
Numerade Educator
02:38

Problem 15

Solve
$$
\frac{d y}{d x}=-\frac{x+y}{3 x+3 y-4}.
$$

Sam Stansfield
Sam Stansfield
Numerade Educator
02:09

Problem 16

If $u=1+\tan y$, calculate $d(\ln u) / d y ;$ hence find the general solution of
$$
\frac{d y}{d x}=\tan x \cos y(\cos y+\sin y).
$$

Keshav Singh
Keshav Singh
Numerade Educator
02:17

Problem 17

Solve
$$
x\left(1-2 x^{2} y\right) \frac{d y}{d x}+y=3 x^{2} y^{2},
$$
given that $y(1)=1 / 2$

M Hassan Anwar
M Hassan Anwar
Numerade Educator
04:17

Problem 18

A reflecting mirror is made in the shape of the surface of revolution generated by revolving the curve $y(x)$ about the $x$-axis. In order that light rays emitted from a point source at the origin are reflected back parallel to the $x$-axis, the curve $y(x)$ must obey
$$
\frac{y}{x}=\frac{2 p}{1-p^{2}}
$$
where $p=d y / d x$. By solving this equation for $x$ find the curve $y(x)$.

Keshav Singh
Keshav Singh
Numerade Educator
02:40

Problem 19

Find the curve such that at each point on it the sum of the intercepts on the $x$ and $y$-axes of the tangent to the curve (taking account of sign) is equal to 1 .

Christy Galilei
Christy Galilei
Numerade Educator
08:18

Problem 20

Find a parametric solution of
$$
x\left(\frac{d y}{d x}\right)^{2}+\frac{d y}{d x}-y=0
$$
as follows.
(a) Write an equation for $y$ in terms of $p=d y / d x$ and show that
$$
p=p^{2}+(2 p x+1) \frac{d p}{d x}
$$
(b) Using $p$ as the independent variable, arrange this as a linear first-order equation for $x$.
(c) Find an appropriate integrating factor to obtain
$$
x=\frac{\ln p-p+c}{(1-p)^{2}}
$$
which, together with the expression for $y$ obtained in (a), gives a parameterisation of the solution.
(d) Reverse the roles of $x$ and $y$ in steps (a) to (c), putting $d x / d y=p^{-1}$, and show that essentially the same parameterisation is obtained.

Keshav Singh
Keshav Singh
Numerade Educator
05:32

Problem 21

Using the substitutions $u=x^{2}$ and $v=y^{2}$, reduce the equation
$$
x y\left(\frac{d y}{d x}\right)^{2}-\left(x^{2}+y^{2}-1\right) \frac{d y}{d x}+x y=0
$$
to Clairaut's form. Hence show that the equation represents a family of conics and the four sides of a square.

Ramon Kryzhan
Ramon Kryzhan
Numerade Educator
04:40

Problem 22

The action of the control mechanism on a particular system for an input $f(t)$ is described, for $t \geq 0$, by the coupled first-order equations:
$$
\begin{aligned}
&\dot{y}+4 z=f(t) \\
&\dot{z}-2 z=\dot{y}+\frac{1}{2} y
\end{aligned}
$$
Use Laplace transforms to find the response $y(t)$ of the system to a unit step input $f(t)=H(t)$, given that $y(0)=1$ and $z(0)=0$.

Keshav Singh
Keshav Singh
Numerade Educator
01:05

Problem 23

Find the general solutions of the following:
(a) $\frac{d y}{d x}+\frac{x y}{a^{2}+x^{2}}=x$;
(b) $\frac{d y}{d x}=\frac{4 y^{2}}{x^{2}}-y^{2}$.

Jacob Steele
Jacob Steele
Numerade Educator
05:58

Problem 24

Solve the following first-order equations for the boundary conditions given:
(a) $y^{\prime}-(y / x)=1, \quad y(1)=-1$
(b) $y^{\prime}-y \tan x=1, \quad y(\pi / 4)=3 ;$
(c) $y^{\prime}-y^{2} / x^{2}=1 / 4, \quad y(1)=1 ;$
(d) $y^{\prime}-y^{2} / x^{2}=1 / 4, \quad y(1)=1 / 2$.

Keshav Singh
Keshav Singh
Numerade Educator
01:59

Problem 25

An electronic system has two inputs, to each of which a constant unit signal is applied, but starting at different times. The equations governing the system thus take the form
$$
\begin{aligned}
&\dot{x}+2 y=H(t) \\
&\dot{y}-2 x=H(t-3).
\end{aligned}
$$
Initially (at $t=0$ ), $x=1$ and $y=0$; find $x(t)$ at later times.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
02:14

Problem 26

Solve the differential equation
$$
\sin x \frac{d y}{d x}+2 y \cos x=1
$$
subject to the boundary condition $y(\pi / 2)=1$.

Keshav Singh
Keshav Singh
Numerade Educator
04:08

Problem 27

Find the complete solution of
$$
\left(\frac{d y}{d x}\right)^{2}-\frac{y}{x} \frac{d y}{d x}+\frac{A}{x}=0,
$$
where $A$ is a positive constant.

Narayan Hari
Narayan Hari
Numerade Educator
04:44

Problem 28

Find the solution of
$$
(5 x+y-7) \frac{d y}{d x}=3(x+y+1).
$$

Keshav Singh
Keshav Singh
Numerade Educator
03:30

Problem 29

Find the solution $y=y(x)$ of
$$
x \frac{d y}{d x}+y-\frac{y^{2}}{x^{3 / 2}}=0,
$$
subject to $y(1)=1$.

Oswaldo JimƩnez
Oswaldo JimƩnez
Numerade Educator
03:20

Problem 30

Find the solution of
$$
(2 \sin y-x) \frac{d y}{d x}=\tan y
$$
if (a) $y(0)=0$, and (b) $y(0)=\pi / 2$.

Keshav Singh
Keshav Singh
Numerade Educator
02:41

Problem 31

Find the family of solutions of
$$
\frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}+\frac{d y}{d x}=0
$$
that satisfy $y(0)=0$

Sam Stansfield
Sam Stansfield
Numerade Educator