"File DIF_APRX.MTH"

"Requires the ODE_APPR.MTH file"

E_(v,i):=ELEMENT(v,i)

D_(v):=DIMENSION(v)

FIRST(r,z,d):=VECTOR(IF(k<d,E_(z,k+2),r),k,d)

"###################"

"USER FUNCTION"

APRX(r,z,z0,h,n):=EXTRACT_2_COLUMNS(RK(FIRST(r,z,D_(z)-1),z,z0,h,n),1,2)

"###############"

"INSTRUCTIONS"

" The APRX(r,z,z0,h,n) function applies n iterations"

"... of the Runge-Kutta method with step size h"

"... to approximate the solution of y^(n)=r(x,y,y',y'',...)"

"... with initial condition z0=[x0,y0,y0',...]"

"The names of y', y'', y''', ... must be replaced by"

"... single letter variable names."

"z is a vector of variable names"

"####################"

"Example"

"To approximate the solution of y'''=(2y'y'')/(2y-1)"

"... y(0)=y'(0)=0, y''(0)=1"

"... first use a to name y' and b to name y''.  y'''=(2ab)/(2y-1)"

"approX the following to get an approximate solution"

"... using 40 steps of length 0.1"

APRX(2*a*b/(2*y-1),[x,y,a,b],[0,0,0,2],0.1,40)

"(The exact solution is y=sin^2 x.  It is fun to"

"... compare the graphs."

"####################"

" IF THIS FILE BEEPED AT YOU WHEN YOU LOADED IT"

"... RELOAD IT IN TWO STEPS."

"First TRANSFER LOAD UTILITY ODE_APPR.MTH"

"Next TRANSFER LOAD DERIVE DIF_APPR.MTH"

"##################"