Lecture Notes in Differential Equations - Bruce E. Shapiro

95
Example 11.1. Construct the Picard iterates of the initial value problem
}
y ′ = −2ty
(11.21)
y(0) = 1
and determine if they converge to the solution.
In terms of equations (11.12) through (11.15), equation (11.21) has
f(t, y) = −2ty (11.22)
t 0 = 0 (11.23)
y 0 = 1 (11.24)
Hence from (11.12)
φ 0 (t) = y 0 = 1 (11.25)
f(s, φ 0 (s)) = −2sφ 0 (s) = −2s (11.26)
φ 1 (t) = y 0 +
= 1 − 2
∫ t
0
∫ t
0
f(s, φ 0 (s))ds (11.27)
sds (11.28)
= 1 − t 2 (11.29)
We then use φ 1 to calculate f(s, φ 1 (s)) and then φ 2 :
f(s, φ 1 (s)) = −2sφ 1 (s) = −2s(1 − s 2 ) (11.30)
φ 2 (t) = y 0 +
= 1 − 2
∫ t
0
∫ t
0
f(s, φ 1 (s))ds (11.31)
s(1 − s 2 )ds (11.32)
= 1 − t 2 + 1 2 t4 (11.33)
Continuing as before, use φ 2 to calculate f(s, φ 2 (s)) and then φ 3 :
f(s, φ 2 (s)) = −2sφ 2 (s) = −2s
(1 − s 2 + 1 )
2 s4
(11.34)
φ 3 (s) = y 0 +
= 1 − 2
∫ t
0
∫ t
0
f(s, φ 2 (s))ds (11.35)
s
(1 − s 2 + 1 )
2 s4 ds (11.36)
= 1 − t 2 + 1 2 t4 − 1 6 t6 (11.37)