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

346 Differential Equations [Ch 9
From the fact that y'=p = when x = 0, we have = ^ 0, i.e., Cj==0.
Hence,
or
___ ____
dx~~ 2 '
whence, integrating once again, w? obtain
Putting y~ when x 0, we find C 2 = 0. Hence, the desired particular
solution is
y^
2) If a differential equation does not contain x explicitly, for instance,
then, putting y'=p, y" p-?-> we get an equation of an order one unit
lower:
Example 2. Find the particular solution of the equation
provided that /=!, #' = when # = 0.
Solution. Put y' = p t then tf^p-- and our equation becomes
We have obtained an equation of the Bernoulli type in p (y is considered
the argument). Solving it, we find
or
From the fact that f/'=p = when r/=l, we have C 1 = 1. Hence,
P=y Vy^
Integrating, we have
* 2 -
arc cos x = C 2 ,
Putting y=l and *=0, we obtain C 2 = 0, whence = cosx or t/==secx.