Lecture Notes in Differential Equations - Bruce E. Shapiro

98 LESSON 11. PICARD ITERATION
Our final step is to verify that the solution found in this way actually works
(because we haven’t actually stated any theorems that say the method
works!). First we observe that
d
dt φ(t) = d dt e−t2 = −2te −t2 = −2tφ(t) (11.57)
which agrees with the first line of equation (11.21). To verify the initial
condition we can calculate
as required.
φ(0) = e −(0)2 = 1 (11.58)