The command dsolve solves ordinary differential equations analytically, although not all ODE's can be solved in closed form. But it is always possible to obtain a series solution which may be useful. We are given the ODE
and the initial condition and we want to find the first
few terms in the series
Write a Maple procedure which on input of and the initial
condition
constructs a linear system of equations to solve.
I.e. let
substitute this finite sum into the ODE, equate coefficients and solve
for the unknowns . Note, you will want to use the taylor command
to truncate the result to order
. Test your Maple procedure on
the following ODE
You should get the following series
Compute the solution analytically using Maple's dsolve command and check that your series solution agrees with Maple's analytic solution.
Develop a Newton iteration to solve for the series which
converges quadratically. I.e. the
'th approximation
is accurate to . Thus the iteration starts with
and computes
then
and so on. Write a Maple procedure to compute the series.