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

.Sec. 16} Integration of Differential Equations by Power Series 361
3091**. A shell leaves a gun with initial velocity u at an
angle a to the horizon. Find the equation of motion if we take
the air resistance as proportional to the velocity.
3092*. A material point is attracted by a centre with a
force proportional to the distance. The motion begins from point A
at a distance a from the centre with initial velocity perpen-
dicular to OA. Find the trajectory.
Sec. 16. Integration of Differential Equations by Means of Power Series
If it is not possible to integrate a differential equation with the help of
elementary functions, then in some cases its solution may be sought in the
form of a power series:
00
y=2 '(* *o) n - 0)
n = o
The undetermined coefficients cn (n = \, 2, ...) are found by putting the
series (1) into the equation and equating the coefficients of identical powers
of the binomial x x on the left-hand and right-hand sides of the resulting
equation.
We can also seek the solution of the equation
in the form of the Taylor's series
y(*) = ^ y^^ (*-*)" (3)
where y(x Q) = yQ , y' (x ) = f (* , t/ ) and the subsequent derivatives y (n) (x )
(n- 2, 3, ...) are successively found by differentiating equation (2) and by
putting X Q in place of x
Example 1. Find the solution of the equation
Solution. We put
y =
whence, differentiating, we get
y" = 2.\Ct + 3.2c sx+...+n(n-\)cn x n -* + (n+l
+ In + 2)(rt-t-l)