reset()
# bepaling mismatch-functie voor -y''=E y, y(0)=y(pi)=0
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x = var('x')
E = var('E')
xmatch = var('xmatch')
y = function('y')(x)
xmatch=pi/2
assume(E>0)
yL=desolve(-diff(y,x,x)==E*y, y, ivar = x,ics = [0,0,1]) # y(0)=0 ; y'(0)=1
yR=desolve(-diff(y,x,x)==E*y, y, ivar = x,ics = [pi,0,1]) # y(pi)=0; y'(pi)=1
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yR=yR.simplify_full().simplify_trig()
yR=yR.trig_reduce()
show(yL)
show(yR)
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# we bekijken het geval waarbij E=4
Evalue = var ('Eval')
Evalue = 4
p1=plot(yL.subs(E=Evalue),x,0,xmatch,color='green')
p2=plot(yR.subs(E=Evalue),x,xmatch,pi,color='red')
p1+p2
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expr = yL*diff(yR,x)-yR*diff(yL,x)
phi=expr.simplify_full().simplify_trig()
show(phi)
p=plot(phi.subs(x=xmatch),E,0,10*pi)
p.show(ymin=-1.5, ymax=2.5)
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