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| Differential Equations Take Home Quiz # 6 with(plots): with(DEtools): |
| Part #1 |
| eq1:=diff(x(t),t) = y(t); eq2:=diff(y(t),t) = -k/m*x(t)-b/m*y(t); ic:[[x(0)=xo,y(0)=vo]]; |
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| Part #2 |
| eqn1:=diff(x(t),t) = y(t); eqn2:=diff(y(t),t) = -1/1*x(t)-0/1*y(t); ic:=[[x(0)=0,y(0)=4]]; vars:=[x(t),y(t)]: p:=dsolve({eqn1,eqn2,x(0)=0,y(0)=4},vars,type=numeric): y1plot:=odeplot(p,[[t,x(t)]],0..10,numpoints=100,color=red,labels=["Time","Position"],title="Position as a Function of Time"): display(y1plot); y2plot:=odeplot(p,[[t,y(t)]],0..10,numpoints=100,color=blue,labels=["Time","Velocity"],title="Velocity as a Function of Time"): display(y2plot); |
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| Part #3 If the equation from part #1 is a pendulum then the graph represented immediately above would be a graph of the pendulum swinging back and forth undampened for an infinite period of time. It starts at at the equilibrium point with an initial velocity of 4 and then travels to the top at 4 and then travels back down down and back into the equilibrium position, but now it is traveling the opposite direction with a velocity again of 4 and then it travels to the top on that side where it has a velocity of zero and then it falls down in the first direction and it reaches its max speed of four at the bottom (equilibrium position) and it carries out this cycle for an infinite period of time as their is no dampening factor. |
| Part #4 |
| eqn1:=diff(x(t),t) = y(t); eqn2:=diff(y(t),t) = -1/1*x(t)-0/1*y(t); ic:=[[x(0)=0,y(0)=4]]; vars:=[x(t),y(t)]: DEplot({eqn1,eqn2},vars,t=0..24,ic,linecolor=[black],stepsize=.01); |
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| Part #5 |
| eqn3:=diff(x(t),t) = y(t); eqn4:=diff(y(t),t) = -1/1*x(t)-(1.5*x^2-1.5)*y(t); ic:=[[x(0)=1,y(0)=0]]; vars:=[x(t),y(t)]: DEplot({eqn3,eqn4},vars,t=0..10,ic,linecolor=[green],title="Dampened Pendulum Phase Plane",stepsize=.01); |
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| Part #4 (Big) |
| eqn1:=diff(x(t),t) = y(t); eqn2:=diff(y(t),t) = -1/1*x(t)-0/1*y(t); ic:=[[x(0)=0,y(0)=4]]; vars:=[x(t),y(t)]: DEplot({eqn1,eqn2},vars,t=0..24,ic,linecolor=[black],title="Undampened Pendulun Phase Plane",stepsize=.01); |
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