11 Ordinary Differential Equations III

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• Solutions must be exact and fully simplified.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

1. Solve
   
   $$y'' + 9 y = t^2 \qquad y(0) = y'(0)= 0$$
   
   
   That would of course be quick using undetermined coefficients. Unfortunately, you must use Laplace transforms. You may only use the brief Laplace transform table handed out in class. Everything else must be derived. Do not use convolution. In solving the system of 5 equations in 5 unknowns of the partial fraction expansion, you may mess around; this is no longer linear algebra. However, you must substitute your solution into the original ODE and ICs and go back to fix any problem there may be.

2. Resonant forcing of an undamped spring-mass system over some time period $T$ that spans a large number of periods can introduce large-amplitude vibrations. To study the problem, consider the example
   
   $$m \ddot x + k x = F(t) \qquad x(0) = \dot x(0) = 0$$
   
   
   where the mass, spring constant, and applied force are given by
   
   $$m=1 \qquad k=4 \qquad F(t)=\cos(2t)\mbox{ if } t<T \qquad F(t)= 0 \mbox{ if } t>T$$
   
   
   Solve using the Laplace transform method. (Note: from S8 and S11 you can see that
   
   $$\sin(\omega t) - \omega t \cos(\omega t) \Longleftrightarrow \frac{2\omega^3}{(s^2+\omega^2)^2}$$
   
   
   Call it result S15.) Clean up your answer. I find that beyond time $t=T$, the amplitude stays constant at
   
   $${\textstyle\frac{1}{4}} T \sqrt{1 + 2\cos(2T)\frac{\sin(2T)}{2T} + \left(\frac{\sin(2T)}{2T}\right)^2}$$
   
   
   which is approximately proportional to $T$ for large $T$. Do your results agree?

3. The generic undamped spring-mass system with external forcing is
   
   $$m \ddot x + k x = F(t) \qquad x(0)=x_0 \quad \dot x(0)=v_0$$
   
   
   where the mass m and spring constant $k$ are given positive constants, $F(t)$ is the given external force, and the initial displacement $x_0$ and velocity $v_0$ are given constants. Give the solution using Laplace transformation, as always restricting use of convolution to the bare minimum. Write the solution in the form
   
   $$A(t) \sin(\omega t) + B(t) \cos(\omega t)$$
   
   
   Identify the natural frequency $\omega$.

   Next check that for a resonant force $F=F_0\cos(\widetilde\omega t)$ with $F_0$ a constant and $\widetilde\omega=\omega$, the integrand in the $A$ integral is always positive, so $A$ will grow without bound in time. However, the integrand in the $B$ integral will periodically change sign, so that $B$ stays finite. Address the question why if I apply a cosine forcing, it is the magnitude of the sine that keeps increasing instead of the cosine. Does that make physical sense? Also verify that for nonresonant forcing, $\widetilde\omega\ne\omega$, both integrands will periodically change sign, so that both $A$ and $B$ stay finite.

4. Solve the system
   $$\begin{eqnarray*} x_1' & = & 2 x_1 + x_2 - 2 x_3 \\ x_2' & = & 3 x_1 -2 x_2 \\ x_3' & = & 3 x_1 + x_2 - 3 x_3 \end{eqnarray*}$$
   
   
   Find the general solution to this system in vector form and in terms of a fundamental matrix. Then find the vector of integration constants assuming that $\vec{x}(0)=(1,7,3)^{\rm T}$ and write $\vec{x}(t)$ for that case.

5. Given the system
   
   $$\dot {\vec x} = A \vec x \qquad A = \left(\begin{array}{cc} 0 & 5 -1 & -2 \end{array} \right)$$
   
   
   Find the general solution to this system in vector form and in terms of a fundamental matrix. Complex solutions not allowed.