If you've done Euler's method or separation of variables, you saw in the first order case how you have to solve for the derivative as a function of x and y. This extends to higher orders... suppose we have a second order differential equation
[tex]\frac{d^2 y}{dx^2} = x*y + \frac{dy}{dx}[/tex]
and y(0)=1, y'(0)=1
I want to do an Euler's method kind of calculation. This requires knowing the derivative at every step. Unfortunately I don't have a formula for the derivative, so I have to use the second derivative to re-calculate the derivative each time
y(.1)=y(0)+.1*y'(0) = 1.1
y'(.1)=y'(0)+.1*y''(0)
I know what y''(0) is from the differential equation: y''(0)=0*1+1=1
y'(.1)=1+.1*1=1.1
Now if I want to update to get y(.2) and y'(.2), I need to know what y''(.1) is. So I use the differential equation
y''(.1) = xy+y' = .1*1.1+1.1 = 1.21
Now I can go to x=.2
y(.2)=y(.1)+.1*y'(.1)=1.21
y'(.2)=y'(.1)+.1*y''(.1)=1.221
Now I need to know what y''(.2) is because I want to be able to calculate y and y' at x=.3, etc. So
y''(.2)=x*y+y' = .2*1.21+1.221 = 1.463
Now I can keep going, estimating y(x) for as large a value of x as I want. The computation only required being able to solve for the second derivative of y at each step, since I already knew all the lower derivatives, so it was useful to have the equation in the form y''=f(x,y,y')
