# Graphic Solution of a Second-Order Differential Equation

Graphic Solution of a Second-Order Differential Equation

This Demonstration shows the Euler–Cauchy method for approximating the solution of an initial value problem with a second-order differential equation. An example of such an equation is , with derivatives from now on always taken with respect to . This equation can be written as a pair of first-order equations, , .

y''+y'+y=0

2

x

x

y'=z

z'=-y+z

2

x

More generally, the method to be described works for any system of two first-order differential equations , with initial conditions , . The particular kinds of systems used as examples here, , reduce to that general type by introducing to get the system , .

y'=f(x,y,z)

z'=g(x,y,z)

y()=

x

1

y

1

z()=

x

1

z

1

y''=g(x,y,y')

z=y'

y'=z

z'=g(x,y,z)

The method consists of simultaneously calculating approximations of (cyan) and (green):

y(x)

z(x)

y

i+1

y

i

x

i

y

i

x

i

y

i

z

i

z

i+1

z

i

x

i

z

i

x

i

y

i

z

i

i=1,….n-1

The pairs are the coordinates of points , , …, that form the so-called Euler's polygonal line that approximates the graph of the function . In the same way, the pairs are the coordinates of points , , …, that form Euler's polygonal line, which approximates the graph of the function .

(,)

x

i

y

i

P

1

P

2

P

n

y(x)

(,)

x

i

z

i

T

1

T

2

T

n

z(x)

The Euler method is the most basic approximation method. The Demonstration compares it with more advanced methods given by the built-in Mathematica function NDSolve.