LAC08: Characteristics MrG 2012.0608

3119 days ago by lac2012

#1) Fave Characteristic Solutions var('r') a(r)=r^2+2*r-1 show(solve(a(r)==0,r)) b(r)=r^2+2*r show(solve(b(r)==0,r)) c(r)=r^2+2*r+1 show(solve(c(r)==0,r)) d(r)=r^2+2*r+2 show(solve(d(r)==0,r)) e(r)=r^2+4 show(solve(e(r)==0,r)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\left[r = -\sqrt{2} - 1, r = \sqrt{2} - 1\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[r = \left(-2\right), r = 0\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[r = \left(-1\right)\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[r = \left(-i - 1\right), r = \left(i - 1\right)\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[r = \left(-2 i\right), r = \left(2 i\right)\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[r = -\sqrt{2} - 1, r = \sqrt{2} - 1\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[r = \left(-2\right), r = 0\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[r = \left(-1\right)\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[r = \left(-i - 1\right), r = \left(i - 1\right)\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[r = \left(-2 i\right), r = \left(2 i\right)\right]
#2) Fave Characteristic Discriminants def disc(a,b,c): return b^2-4*a*c print "disc(1,2,-1) =",disc(1,2,-1) print "disc(1,2,0) =",disc(1,2,0) print "disc(1,2,1) =",disc(1,2,1) print "disc(1,2,2) =",disc(1,2,2) print "disc(1,0,4) =",disc(1,0,4) 
       
disc(1,2,-1) = 8
disc(1,2,0) = 4
disc(1,2,1) = 0
disc(1,2,2) = -4
disc(1,0,4) = -16
disc(1,2,-1) = 8
disc(1,2,0) = 4
disc(1,2,1) = 0
disc(1,2,2) = -4
disc(1,0,4) = -16
#3) Fave Characteristic Graphs pa=plot(a(r),-3,3,color='red') pb=plot(b(r),-3,3,color='green') pc=plot(c(r),-3,3,color='cyan') pd=plot(d(r),-3,3,color='magenta') pe=plot(e(r),-3,3,color='chartreuse') show(pa+pb+pc+pd+pe) 
       
#4) Fave DiffEqu Solutions var('t') x=function('x',t) show(desolve(diff(x,t,2)+2*diff(x,t,1)-x,x)) show(desolve(diff(x,t,2)+2*diff(x,t,1),x)) show(desolve(diff(x,t,2)+2*diff(x,t,1)+x,x)) show(desolve(diff(x,t,2)+2*diff(x,t,1)+2*x,x)) show(desolve(diff(x,t,2)+4*x,x)) 
       
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#5) R,!Q,!= OverDamped aka nonHarmonic Disc>0 show(desolve(diff(x,t,2)+2*diff(x,t,1)-x,x,[0,2,0])) show(plot(desolve(diff(x,t,2)+2*diff(x,t,1)-x,x,[0,2,0]),0,10)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{2} \, {\left(\sqrt{2} - 2\right)} e^{\left(-{\left(\sqrt{2} + 1\right)} t\right)} + \frac{1}{2} \, {\left(\sqrt{2} + 2\right)} e^{\left({\left(\sqrt{2} - 1\right)} t\right)}
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#6) R,Q,!= OverDamped aka nonHarmonic Disc>0 show(desolve(diff(x,t,2)+2*diff(x,t,1),x,[0,2,1])) show(plot(desolve(diff(x,t,2)+2*diff(x,t,1),x,[0,2,1]),0,10)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{2} \, e^{\left(-2 \, t\right)} + \frac{5}{2}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{2} \, e^{\left(-2 \, t\right)} + \frac{5}{2}
#7) R,Q,= Critically Damped Harmonic Disc=0 show(desolve(diff(x,t,2)+2*diff(x,t,1)+x,x,[0,2,0])) show(plot(desolve(diff(x,t,2)+2*diff(x,t,1)+x,x,[0,2,0]),0,10)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}2 \, {\left(t + 1\right)} e^{\left(-t\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}2 \, {\left(t + 1\right)} e^{\left(-t\right)}
#8) A+Bi Under Damped Harmonic Disc<0 show(desolve(diff(x,t,2)+2*diff(x,t,1)+2*x,x,[0,2,0])) show(plot(desolve(diff(x,t,2)+2*diff(x,t,1)+2*x,x,[0,2,0]),0,10)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}2 \, {\left(\sin\left(t\right) + \cos\left(t\right)\right)} e^{\left(-t\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}2 \, {\left(\sin\left(t\right) + \cos\left(t\right)\right)} e^{\left(-t\right)}
#9) 0+Bi UnDamped Harmonic aka Simple Harmonic Disc<0 show(desolve(diff(x,t,2)+4*x,x,[0,2,0])) show(plot(desolve(diff(x,t,2)+4*x,x,[0,2,0]),0,10)) 
       
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