Mathematical Methods for Physics and Engineering : A Comprehensive Guide

FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS

Find an expression for the velocityvof the mass as a function of time, given
that it has an initial velocityv 0.
14.13 Using the results about Laplace transforms given in chapter 13 fordf/dtand
tf(t), show, for a functiony(t)thatsatisfies

t

dy

dt

+(t−1)y=0 (∗)

withy(0) finite, that ̄y(s)=C(1 +s)−^2 for some constantC.

Given that

y(t)=t+

∑∞

n=2

antn,

determineCand show thatan=(−1)n−^1 /(n−1)!. Compare this result with that
obtained by integrating (∗)directly.
14.14 Solve
dy
dx

=

1

x+2y+1

.

14.15 Solve
dy
dx

=−

x+y

3 x+3y− 4

.

14.16 Ifu=1+tany,calculated(lnu)/dy; hence find the general solution of

dy

dx

=tanxcosy(cosy+siny).

14.17 Solve

x(1− 2 x^2 y)

dy

dx

+y=3x^2 y^2 ,

given thaty(1) = 1/2.
14.18 A reflecting mirror is made in the shape of the surface of revolution generated by
revolving the curvey(x) about thex-axis. In order that light rays emitted from a
point source at the origin are reflected back parallel to thex-axis, the curvey(x)
must obey
y
x

=

2 p

1 −p^2

,

wherep=dy/dx. By solving this equation forx, find the curvey(x).
14.19 Find the curve with the property that at each point on it the sum of the intercepts
on thex-andy-axes of the tangent to the curve (taking account of sign) is equal
to 1.
14.20 Find a parametric solution of

x

(

dy

dx

) 2

+

dy

dx

−y=0

as follows.

(a) Write an equation foryin terms ofp=dy/dxand show that

p=p^2 +(2px+1)

dp

dx

.

(b) Usingpas the independent variable, arrange this as a linear first-order

equation forx.