# Question:How do I numerically solve differential equations?

## Question:How do I numerically solve differential equations?

Maple
None of the built-in features of Maple will help me numerically solve the kinds of complicated differential equations I want. I was hoping that Maple's option in dsolve of solving by Taylor series would at least grind out the first nine terms of the Taylor series expansion of the solution y(x) of e.g. x^3 + (y'(x) - 9*x)^(5/(y(x)+2)) + 4*(x-y(x))^(y'(6*x+4)) = 0 subject to y(1)=0. I dumped in functional equations, too, to see if Maple could handle it. Ok. So, for something like this, R(x,y(x),y'(x),y'(6x+4))=0 Maple should be at least able to do the first 9 differentiations to express y''(x) through y^(10)(x) in terms of x, y(x), y'(x), y'(6x+4). Then, when it gets to the initial condition, y(1)=0, y'(1) and y'(6*1+4)=y'(10) are fixed by 1 + (y'(1) - 9)^(5/2) + 4*(1)^(y'(10)) = 0 so then Maple should be able to tell me: can't go further. Actually, in this example, it can: y'(1)= 34, since (34-9)^(5/2) = -5 = - 4*(1)^(y'(10)) -1 no matter what value y'(10) happens to be. So, I will need help writing my own routine to symbolically differentiate any differential equation I give Maple, solve for the higher order derivatives in terms of the lower order ones, and plug in initial (or boundary value) conditions. Any help will be appreciated. Thanks, John
﻿