Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

356_Differential Equations_(Ch. 9
3042. A material point of mass m is attracted by each of two
centres with a force proportional to the distance (the constant
of proportionality is k). Find the law of motion of the point
knowing that the distance between the centres is 26, at the initial
instant the point was located on the line connecting the
centres (at a distance c from its midpoint) and had a velocity
of zero.
3043. A chain of length 6 metres is sliding from a support
without friction. If the motion begins when 1 m of the chain
is hanging from the support, how long will it take for the entire
chain to slide down?
3044*. A long narrow tube is revolving with constant angular
velocity o> about a vertical axis perpendicular to it. A ball inside
the tube is sliding along it without friction. Find the law
of motion of the ball relative to the tube, considering that
a) at the initial instant the ball was at a distance a from
the axis of rotation; the initial velocity of the ball was zero;
b) at the initial instant the ball was located on the axis of
rotation and had an initial velocity v 9 .
Sec. 13. Linear Differential Equations of Order Higher than Two with
Constant Coefficients
#t
1. Homogeneous equations. The fundamental system of solutions
f Un a homogeneous
y lt
linear equation with constant coefficients
y< n > < n
+ aiy - + . . . + an _,y' +any = (1)
is constructed on the basis of the character of the roots of the characteristic
equation
Q. (2)
Namely, 1) if k is a real root of the equation (2) of multiplicity m, then to
this root there correspond m linearly independent solutions of equation (1):
2) if a i p/ is a pair of complex roots of equation (2) of multiplicity m,
then to the latter there correspond 2m linearly independent solutions of
equation (1):
y l = e *x cos PX, #, = e* x sin px, y t = xe* x cos PX, y4 = xe* x sin PX, ...
2. Inhotnogeneous equations. A particular solution of the inhomogeneous
equation
is sought on the basis of rules 2 and 3 of Sec. 12.
(3)