#
# Solutions to a couple of quiz problems in Maple.  The first uses the
# Wronskian and the second dsolve with type=numeric
#
> restart;
# The first question requires no Maple. The issue is what happens to the 
# Wronskian of  
# two solutions to y''+x*y'+x^2*y=0 as x->Infinity. Well, here p=x, so 
# W'=-x*W so  
# W=c*exp(-int(p(x),x)=c*exp(-x^2/2). This goes to zero as x->Infinity.
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> 
# The second question can be worked by hand or by computer. The computer 
# solution is only numeric. You need to (still) be conversant with dsolve. 
# Naturally, you call up the syntax rulebook:
> ?dsolve
# EXAMPLES:   
# > dsolve(diff(y(x),x$2) - y(x) = 1, y(x));
#                      y(x) = - 1 + _C1 exp(x) + _C2 exp(- x)
#    
# > dsolve({diff(v(t),t)+2*t=0, v(1)=5}, v(t));
#                                           2
#                                 v(t) = - t  + 6
#    
> 
# Next, you write out the DE and some initial conditions.
> dq:=diff(y(x),x$2)+x*diff(y(x),x)+x^2*y(x)=0;

                      /   2      \
                      |  d       |     /  d      \    2
                dq := |----- y(x)| + x |---- y(x)| + x  y(x) = 0
                      |   2      |     \ dx      /
                      \ dx       /
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> inits:=y(1)=1,D(y)(1)=0;

                         inits := y(1) = 1, D(y)(1) = 0
> answer:=dsolve({dq,inits},y(x),numeric);

answer := proc(rkf45_x) ... end

> answer(2);

                                            d
        [x = 2, y(x) = .4741157266787145, ---- y(x) = -.9013535642279813]
                                           dx
# Now the problem is how to display the results. This code string below 
# picks out the right hand side of the second entry in answer. That's 
# because the second entry is the formula for y itself. The object formed 
# is a list of points, with the first entry to each  
# point being 0.02*k=x, and the second, the corresponding value of y.
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> yvals:=[seq([.02*k,rhs(answer(.02*k)[2])],k=1..200)]:
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> plot(yvals);
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> answer2:=dsolve({dq,y(1)=0,D(y)(1)=1},y(x),numeric);

answer2 := proc(rkf45_x) ... end

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> yvals2:=[seq([.02*k,rhs(answer2(.02*k)[2])],k=1..200)]:
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> with(plots):p1:=plot(yvals):p2:=plot(yvals2):display({p1,p2});
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> 

   ** Maple V Graphics **

# The two plots have interwoven zeros. As x increases, they're both 
# crashing toward zero. This was foreshadowed by the hand part of the 
# quiz, in which the Wronskian was called for. The Wronskian of two 
# solutions to the DE y''+xy'+x^2y=0 has the  
# property that W[y1,y2](x)=c*exp(-(1/2)*x^2). This negative squared 
# exponential,  however, drops (at first slowly, then with great speed), 
# toward zero. So we should expect that the solutions will either become 
# close to zero, or at least close to each other, as x increases.
> answer3:=dsolve({dq,y(0)=1,D(y)(0)=0},y(x),numeric);

answer3 := proc(rkf45_x) ... end

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> yvals3:=[seq([.02*k,rhs(answer3(.02*k)[2])],k=1..200)]:\
p3:=plot(yvals3):display(p3);
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>