University:
Whitman College
Course:
MATH-244 | Differential Equations
Academic year:

2020
Views:

130

Pages:

3
Author:

Kajbahe4b

Power Series solutions in Maple The relevant Maple file is a separate document, and the output to the Maple document is also available as a separate PDF. A power series solution to an ODE may be obtained in one of two different ways: • Using powseries package and powsolve – This gives a procedure from which a truncated series of any order can be produced useing tpsform – Power series is always based at 0. – You can extract the recurrence relation directly. – Cannot use non-polynomial coefficients • Using dsolve with a ’series’ option – Series is based at initial condition. – Can use non-polynomial coefficients, non-polynomial forcing. – Cannot extract the recurrence relation directly. In both cases, you can transform a series into a truncated polynomial (for plotting, etc) by using convert Examples using powseries 1. (See PS1.mws) Solve, using a power series: y 00 + xy 0 + 2y = 0, y(0) = a0 , y 0 (0) = a1 with(powseries): deq:=diff(y(x), x$2) + x*diff(y(x), x) + 2*y(x) = 0; inits:=y(0)=a[0],D(y)(0)=a[1]; IVP:={deqn,inits}; #The following computes the series solution # f is a procedure, F is the series f:=powsolve(IVP); F:=tpsform(f,x,12); #We can extract the recursion: 1 f(_k); #or, more succinctly: recursion_relation:=a(n)=subs(_k=n,f(_k)); 2. (See PS2.mws) Let y 00 − (3x − 2)y 0 − 2y = 0, y(0) = 0, y 0 (0) = 2 (a) Find the general power series solution using Maple. (b) Find the recurrence relation. (c) Compute the order 7 and order 13 approximations. (d) Graphically compare the two previous solutions, together with Maple’s default solution. (e) Is there an x value for which the solutions suddenly stop matching? See if you can get a good approximation. unassign(’y’); eqn:=diff(y(x),x$2)-(3*x-2)*diff(y(x),x)-2*y(x)=0; inits:=y(0)=0,D(y)(0)=2; IVP:={eqn,inits}; with(powseries): f:=powsolve(IVP); recursion_relation:=a(n)=subs(_k=n,f(_k)); f6:=tpsform(f,x,7); f12:=tpsform(f,x,13); F6:=convert(f6,polynom,x); F12:=convert(f12,polynom,x); g:=dsolve(IVP,y(x)); #This will give Maple’s default solution G:=rhs(g); plot({G,F6,F12},x=-1..1,y=-2..2,numpoints=150); Examples using dsolve 1. (See DS1.mws) Solve, using a power series to order 12: y 00 + xy 0 + 2y = sin(x), 2 y(1) = a0 , y 0 (1) = a1 unassign(’y’); Order:=12; deq:=diff(y(x), x$2) + x*diff(y(x), x) + 2*y(x) = sin(x); inits:=y(1)=a[0],D(y)(1)=a[1]; IVP:={deq,inits}; F:=dsolve(IVP,y(x),’series’); # Here is how you convert this into a regular polynomial # (useful for plotting, if we had numbers in the initial # conditions). convert(rhs(F),polynom,x); 2. (See Airy1.mws) Construct a picture like Figure 5.2.4, pg. 245 in Boyce and Diprima. That is, construct the partial power series solutions to Airy’s equation, y 00 − xy = 0, y(0) = 0, y 0 (0) = 1 We follow the same basic procedures as before. In this case, we could use powsolve, but we’ll go ahead and use dsolve unassign(’y’); deq:=diff(y(x),x$2)-x*y(x)=0; inits:=y(0)=0,D(y)(0)=1; IVP:={deqn,inits}; Order:=5; #Make this one bigger than the poly degree f4:=dsolve(IVP,y(x),’series’); F4:=convert(rhs(f4),polynom,x); #Now repeat those last two lines, changing the order # to create F4, F10, F16, F22 #And plot them: plot({F4,F10,F16,F22},x=-10..2,y=-3..3); 3

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