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:
dy
 f ( x, y )
dx
,
y ( x0 )  y 0
x1
x1
dy
dx

f
(
x
,
y
)
dx
x dx
x
0
0
x1
y ( x1 )  y ( x0 )   dy 
x0
x

f (t , y (t )) dt
x0
x
y ( x)  y ( x0 ) 

x0
f (t , y (t )) dt
Iteration:
x
1 ( x)  y ( x0 )   f (t ,  0 (t )) dt
x0
Nth term:
x
 n 1 ( x)  y ( x0 )   f (t ,  n (t )) dt
x0
Let’s do one.
Set Phasers to Fun!
dy
 y, y(0)  1
dx
f ( x, y)  y
x
x
0
0
1 ( x)  1   0 (t )dt  1   1dt  1  x
x
x
2
x
2 ( x)  1   1 (t )dt  1   1  tdt  1  x 
2
0
0
x
x
t2
x2 x3
3 ( x)  1   2 (t )dt  1   1  t  dt  1  x  
2
2
6
0
0
Nth Term:
2
3
4
n
x
x
x
x
 n ( x)  1  x 


 ... 
2! 3! 4!
n!
But!
This is the Mclaurin Series
Expansion for…
Drum Roll, Please.
(Pause for Dramatic effect…)
e
x
We can check the solution by
using separation of variables
dy
 y, y(0)  1
dx
dy
 dx
y
1
dy

dx
y

ln(y)  x  c
ln( y )
x c
e
e
x
y  ke
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