Solution of Exam on Ordinary Differential Equations. Trial Aleph ...
Solution of Exam on Ordinary Differential Equations. Trial Aleph ...
Solution of Exam on Ordinary Differential Equations. Trial Aleph ...
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P 1<br />
n=0 (n (n 1) a n + 2 (n + 2) (n + 1) a n+2 + 2na n + 3 (n + 1) a n+1 + a n ) t n =<br />
0<br />
P 1<br />
n=0 (2 (n + 2) (n + 1) a n+2 + 3 (n + 1) a n+1 + (n (n 1) + 2n + 1) a n ) t n =<br />
0<br />
2 (n + 2) (n + 1) a n+2 +3 (n + 1) a n+1 +(n (n 1) + 2n + 1) a n = 0; n = 0; 1; 2; :::<br />
(58)<br />
a n+2 =<br />
3<br />
2 (n + 2) a n+1<br />
n 2 + n + 1<br />
2 (n + 2) (n + 1) a n; n = 0; 1; 2; ::: (59)<br />
z (0) = a 0 = 1; z 0 (0) = a 1 = 1<br />
n = 0 :<br />
a 2 = 3 4 a 1<br />
1 4 a 0 = 1<br />
n = 1 :<br />
a 3 = 3 6 a 2<br />
n = 2 :<br />
a 4 = 3 8 a 3<br />
3<br />
12 a 1 = 1 2 ( 1) 1<br />
4 = 1 4<br />
7<br />
24 a 2 = 3 8 1 4<br />
7<br />
19<br />
24<br />
( 1) =<br />
96<br />
n = 3 :<br />
a 5 =<br />
3<br />
2(3+2) a 4<br />
9+3+1<br />
2(3+2)(3+1) a 3 = 3<br />
10<br />
<br />
19<br />
96<br />
13<br />
40 1 4 = 9<br />
64<br />
n = 4 :<br />
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::<br />
z (t) = a 0 +a 1 t+a 2 t 2 +a 3 t 3 +a 4 t 4 +a 5 t 5 +::: = 1+t t 2 + 1 4 t3 + 19<br />
y (x) = z (x 1)<br />
8.2 Answer<br />
96 t4 9<br />
64 t5 +:::;<br />
y (x) = 1 + (x 1) (x 1) 2 + 1 4 (x 1)3 + 19<br />
96 (x 1)4 9<br />
64 (x 1)5 + :::<br />
11