Differential Equations 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: General Form: ,

Iteration: Nth term:

Let’s do one. Set Phasers to Fun!

Nth Term:

This is the Mclaurin Series But! This is the Mclaurin Series Expansion for… Drum Roll, Please.

(Pause for Dramatic effect…)

We can check the solution by using separation of variables

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: “We must anticipate, and not make the same mistake once” -Captain Jean-Luc Picard of the USS Enterprise