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The position of a certain undamped spring–mass system satisfies the initial value problem

$$u''+2u=0$$ $$u(0)=0$$ $$u'(0)=2$$

(a) Find the solution of this initial value problem

(b) Plot $u'$ versus $u$; that is, plot $u(t)$ and $u'(t)$ parametrically with $t$ as the parameter. This plot is known as a phase plot and the $uu'$-plane is called the phase plane. Observe that a closed curve in the phase plane corresponds to a periodic solution $u(t)$. What is the direction of motion on the phase plot as $t$ increases?

What i tried

(a) i am able to solve the ODE for part (a) where i got $$u(t)=2/\sqrt{2}sin\sqrt{2}$$ as my solution. Its the part (b) that im confused as i dont really get what the question mean and how to approach it. I know how to find $u'(t)$ but what should i do after finding $u'(t)$.Does it mean that $u'$ is the first parametric equation and $u$ is the second parametric eqn? Could anyone please explain. Thanks

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  • $\begingroup$ $u(t)=2\sqrt{2}\ \sin(\sqrt{2}t)$ $\endgroup$ – Demosthene Mar 2 '15 at 13:55
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We will start by plotting the solution and the derivative, so we have:

$$u(t) = \sqrt{2} \sin \left(\sqrt{2} t\right), u'(t) = 2 \cos \left(\sqrt{2} t\right)$$

enter image description here

If we parametrically plot ($u(t)$ versus $u'(t)$ with $t$ being the vertical axis) this for a single initial condition, we get an ellipse as:

enter image description here

If we do a general phase portrait, we should get a bunch of ellipses for varying initial conditions (you can see the direction of motion) and this looks like:

enter image description here

You can try this by hand using Parmateric Equations or Parametric Equations and also using many tools like Wolfram Alpha or Graph Sketch.

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Question b is asking you to make a plot with $u'$ on the vertical axis and $u$ on the horizontal axis.

The range is $t=[0;+\infty)$.

How to make such a plot: for each successive value of $t$, starting at $t=0$, compute $u'(t)$ and $u(t)$, and simply place the point $(u,u')$ on the graph.

You should get an ellipse, i.e. after some interval $T$, the period of oscillation, the same values $(u,u')$ are repeated: this is the definition of oscillatory motion.

(If you manage to plot the graph, you can share it and we can discuss it if you want).

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