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

362 Differential Equations [Ch. 9
have
Generally,
Consequently,
(X*
where c =f/ and c l = y'Q .
c s = ^~ and so forth.
2 = (k =1,2, 3, ...).
3.4-6. 7-... -
X1 1 X^ \
* + + + '" + + '"
3T4 3.4.6.7 3.4-6.?.. . .-3* (3fc+ 1) )*
Applying d'Alembert's test, it is readily seen that series (4) converges
for oo < x < + oo .
Example 2. Find the solution of the equation
Solution. We put
y'
We have y =\, i^ = + l = l. Differentiating equation y' = x + y, we succes-
sively find /=! + (/', ^=1 + 1 = 2, y'"=y\ /o" = 2, etc. Consequently,
For the example at hand, this solution may be written in final form as
*-l x) or = 2e* 1 x.
The procedure is similar for differential equations of higher orders. Testing
the resulting series for convergence is, generally speaking,
and is not obligatory when solving the problems of this section.
complicated
With the help of power series, find the solutions of the equations
for the indicated initial conditions.
In Examples 3097, 3098, 3099, 3101, test the solutions
obtained for convergence.
3093. y' = y + x 2
\ y = 2 for * = 0.
3094. y' = 2y + x 1; y = y for x=l.
3095. 0' = / + *; f/ = y
for x==a
3096. y' = x* y*\ # = for * = 0.
3097. (1 x)y' = l+x y\ y = for
(4)