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.