Picard’s Method For Solving

Differential Equations

Not this Picard.

This Picard.

Picard’s Method is an alternative method for finding a solution to differential equations. It uses successive approximation in order to estimate what a solution would look like. The approximations resemble Taylor-Series expansions. As with Taylor-Series when taken to infinity, they cease to be approximations and become the function they are approximating.

What We Will Do

• Derive Picard’s Method in a general form

• Apply Picard’s Method to a simple differential equation.

• Briefly Mention the greater implications and uses of Picard’s Method.

Make it so

Differential Equation:

),( yxfdx

dy

00 )( yxy

,

General Form:

1

0

1

0

),(x

x

x

x

dxyxfdxdx

dy

1

0 0

))(,()()( 01

x

x

x

x

dttytfdyxyxy

x

x

dttytfxyxy0

))(,()()( 0

Iteration:

x

x

dtttfxyx0

))(,()()( 001

x

x

nn dtttfxyx0

))(,()()( 01

Nth term:

Let’s do one.Set Phasers to Fun!

,ydx

dy 1)0( y

yyxf ),(

x x

xdtdttx0 0

01 111)(1)(

x x x

xtdtdttx0

2

0

12 2111)(1)(

621

211)(1)(

3

0

2

0

2

23

xxxdt

ttdttx

x x

!...

!4!3!21)(

432

n

xxxxxx

n

n

Nth Term:

But!

This is the Mclaurin SeriesExpansion for…

Drum Roll, Please.

(Pause for Dramatic effect…)

xe

We can check the solution by using separation of variables

,ydx

dy 1)0( y

dxy

dy

dxdyy

1

cxy )ln(cxy ee )ln(

xkey Boom.

Other Implementations:

Picard’s method is integral (ha ha ha…) to the Picard-Lindeloef Theorem of Existence and Uniqueness of Solutions to Differential Equations. It uses the fact that these successive integral approximations converge which allows you to claim that for a certain region that the solution is unique. Sweet!

But is it useful as a solving method?

This is still unclear. As with most mathematics, you must be able to analyze a problem before you start and decide for yourself what will be the most effective. While it can be a straightforward approach it also gets computationally heavy.

Lastly…

Picard’s Advice on Problem solving:

-Captain Jean-Luc Picard of the USS Enterprise

“We must anticipate, and not make the same mistake once”