Graphic Solution of a First-Order Differential Equation

f

1

initial values

x

1

0.1

y

1

0.3

step

h

0.2

number of points

n

3

show 'exact' solution

show explanation

	1	2	3
x	0.1	0.3	0.5
y	0.3	0.42	0.588
′ y	0.6	0.84	0

This Demonstration presents Euler's method for the approximate (or graphics) solution of a first-order differential equation with initial condition

y'=f(x,y)

,

y(

x

1

)=

y

1

.

The method consists of calculating the approximation of

y(x)

by

x

i+1

=

x

i

+h

,

y

i+1

=

y

i

+hy'(

x

i

)=

y

i

+hf(

x

i

,

y

i

)

,

where

i=1,…,n-1

.

These coordinates determine points

P

1

,

P

2

, …,

P

n

. These points form Euler's polygonal line that is an approximate solution of the problem. The Demonstration compares it with a better solution provided by Mathematica's built-in NDSolve function (brown line).