CK-12-Calculus

http://www.ck12.org Chapter 8. Infinite Series

Yn(x) =y 0 +∫xx 0 f(t,Yn− 1 (t))dtforn≥ 0
The sequence of approximations converges to the solutiony(x), i.e.

nlim→∞Yn(x) =y(x).

Now that we have defined Picard’s method, let’s calculate a sequence of functions for an initial value problem.

Example 1

Find the first four functions{Yn(x)}^3 n= 0 defined by Picard’s method for the solution to the initial value problem

y′(x) =xy(x) withy(− 1 ) =1.

Solution

We want to apply the Fundamental Theorem of Calculus to the differential equations so that it is reformulated for

use in the Picard method. Thus,

∫x

− 1 y

′(t)dt=∫x

− 1 ty(t)dt

y(x)−y(− 1 ) =

∫x

− 1 ty(t)dt

y(x) = 1 +

∫x

− 1 ty(t)dt

Now that the differential equation has been rewritten for Picard’s method, we begin the calculations for the sequence

of functions. In all cases the first functionY 0 (x)is given by the initial condition:

Step 1 - DefineY 0 (x) = 1

Step 2 - SubstituteY 0 (x) =1 fory(t)in the integrand ofy(x) = 1 +∫−x 1 ty(t)dt:

Y 1 (x) = 1 +

∫x

− 1 tdt

Y 1 (x) = 1 +t

2

2

∣∣

∣∣

∣

x

− 1

Y 1 (x) =^12 +x

2

2

Step 3 - SubstituteY 1 (x) =^12 +x 22 fory(t)in the integrand as above:

Y 2 (x) = 1 +

∫x

− 1 t

( 1

2 +

t^2

2

)

dt

Y 2 (x) = 1 +

(t 2

4 +

t^4

8

)∣∣∣

∣∣

x

− 1

Y 2 (x) =^58 +x

2

4 +

x^4

8

Step 4 - SubstituteY 2 (x) = 85 +x 42 +x 84 fory(t)in the integrand as done previously: