Differential Equations -- HomeworkLab 2.3: The Harmonic Oscillator with Modified Dampling restart; with(DEtools): In this report, we discuss the most classic of all second-order equations, the harmonic oscillator. The harmonic oscillator is an example of a second-order, homogeneous, linear equation with constant coefficients. This equation is used to model the motion of a spring. We will describe the motion of the spring assuming certain values of m, b, and k. The value we pick is: m=5, k=5, b=4, a=3. Case 1: In case 1, we consider the harmonic oscillator with no damping, that is b = 0 and with k \007= 0. Examine solutions using both their graphs and the phase plane. Are the solutions periodic? If so, what does the period seem to be? Describe the behavior of three different solutions that have especially different initial conditions and be specific about the physical interpretation of the different initial conditions. (Analytic methods to answer these questions are discussed in Chapter 3. For now, work numerically.) dxdt1 := diff(x(t),t) = 2*x(t); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZkeGR0MUdGKC8tSSVkaWZmR0YmNiQtSSJ4R0YoNiNJInRHRihGNSwkRjIiIiM3I0Yu dydt1 := diff(y(t),t) = -1*y(t); DEplot({dxdt1,dydt1},[x(t),y(t)],t=0..2.5,x=-3..3,y=-3..3); DEplot({dxdt1,dydt1},[x(t),y(t)],t=0..2.5,x=-3..3,y=-3..3, [[x(0)=0,y(0)=3],[x(0)=0,y(0)=-3], [x(0)=0.5,y(0)=0],[x(0)=-0.5,y(0)=0]],linecolor=blue); DEplot({dxdt1,dydt1},[x(t),y(t)],t=0..2.5,x=-3..3,y=-3..3, [[x(0)=0.1,y(0)=3],[x(0)=-0.1,y(0)=3], [x(0)=-0.1,y(0)=-3],[x(0)=0.1,y(0)=-3]],linecolor=blue); And now, Example 2: dxdt2 := diff(x(t),t) = -2*x(t); dydt2 := diff(y(t),t) = 0.5*x(t)-3*y(t); DEplot({dxdt2,dydt2},[x(t),y(t)],t=0..2.5,x=-3..3,y=-3..3); DEplot({dxdt2,dydt2},[x(t),y(t)],t=0..2.5,x=-3..3,y=-3..3, [[x(0)=0,y(0)=3],[x(0)=0,y(0)=-3], [x(0)=3,y(0)=1.5],[x(0)=-3,y(0)=-1.5]],linecolor=blue); DEplot({dxdt2,dydt2},[x(t),y(t)],t=0..2.5,x=-3..3,y=-3..3, [[x(0)=0,y(0)=3],[x(0)=0,y(0)=-3], [x(0)=3,y(0)=1.5],[x(0)=-3,y(0)=-1.5], [x(0)=2,y(0)=3],[x(0)=-2,y(0)=1], [x(0)=-1.5,y(0)=-3],[x(0)=2,y(0)=-2]],linecolor=blue); LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=