Mathematical Methods for Physics and Engineering : A Comprehensive Guide

14.4 EXERCISES

14.5 By finding suitable integrating factors, solve the following equations:

(a) (1−x^2 )y′+2xy=(1−x^2 )^3 /^2 ;

(b)y′−ycotx+cosecx=0;

(c) (x+y^3 )y′=y(treatyas the independent variable).

14.6 By finding an appropriate integrating factor, solve

dy

dx

=−

2 x^2 +y^2 +x

xy

.

14.7 Find, in the form of an integral, the solution of the equation

α

dy

dt

+y=f(t)

for a general functionf(t). Find the specific solutions for
(a) f(t)=H(t),
(b)f(t)=δ(t),
(c) f(t)=β−^1 e−t/βH(t)withβ<α.
For case (c), what happens ifβ→0?
14.8 A series electric circuit contains a resistanceR, a capacitanceCand a battery
supplying a time-varying electromotive forceV(t). The chargeqon the capacitor
therefore obeys the equation

R

dq

dt

+

q

C

=V(t).

Assuming that initially there is no charge on the capacitor, and given that
V(t)=V 0 sinωt, find the charge on the capacitor as a function of time.
14.9 Using tangential–polar coordinates (see exercise 2.20), consider a particle of mass
mmoving under the influence of a forcefdirected towards the origin 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=−mv

dv

dr

and mv^2 =fp

dr

dp

.

Show further thath=mpvis a constant of the motion and that the law of force

can be deduced from

f=

h^2

mp^3

dp

dr

.

14.10 Use the result of exercise 14.9 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 radiusathat passes through the origin;

(b) An equiangular spiral, which is defined by the property that the angleα

between the tangent and the radius vector is constant along the curve.

14.11 Solve

(y−x)

dy

dx

+2x+3y=0.

14.12 A massmis accelerated by a time-varying forceαexp(−βt)v^3 ,wherevis its
velocity. It also experiences a resistive forceηv,whereηis a constant, owing to
its motion through the air. The equation of motion of the mass is therefore

m

dv

dt

=αexp(−βt)v^3 −ηv.