1. Assume that $x_1(t)$ and $x_2(t)$ are (individually) solutions to the following ordinary, second-order differential equations. For which of the following is the linear combination
   
   $$x(t) = c_1 x_1(t) + c_2 x_2(t)$$
   
   
   also a solution?
   1. $\ddot 2 x + \dot x + 5 x = 0$.
   2. $\ddot x + 2 \sin t \dot x + 6 x = 0$.
   3. $\ddot x + 44 \dot x + x = 0$.
   4. $\ddot x + 5 \dot x + 4 x = 3 t$.
   What are the differences between the equations for which $x(t)$ is a solution and $x(t)$ is not a solution?

2. In the case of repeated roots of the characteristic equation
   
   $$\lambda^2 + 2 \zeta \omega_n \lambda + \omega_n^2 = 0,$$
   
   
   prove the following two facts.
   1. If the root is repeated, the value of the root is $\lambda = -\omega_n$.
   2. If the root is repeated, then the two solutions
      
      $$x_1(t) = e^{-\omega t}, \quad x_2(t) = t e^{-\omega t}$$
      
      
      are linearly independent.

3. Determine the solution to $12 \ddot x - 10 \dot x + 2 x = 0$, where $x(0) = 4$ and $\dot x(0) = 0.$

4. This chapter mainly deals with constant-coefficient second-order ordinary differential equations. However, there is one class of variable-coefficient equations that is easy to solve. The equation
   

   $$t^2 \ddot x + \alpha t \dot x + \beta x = 0$$ (1)

   
   
   is called Euler's equation. Show that $x(t) = t^\lambda$ is a solution.
   1. Are there usually two solutions to Euler's equation? If so, are they linearly independent? If they are not linearly independent everywhere, on what intervals do they form a fundamental set of solutions?
   2. Determine the general solution to
      
      $$t^2 \ddot x + 4 t \dot x + 2 x = 0.$$
      
      

5. The following figure illustrates the solution to
   
   $$3 \ddot x + \dot x + 11 x = 0,$$
   
   
   where $x(0) = 1$ and $\dot x(0) = 0$.
   
   By referring to one of the forms of the solution given in Section 3.3, without solving the equation sketch what the solution will look like if
   1. The coefficient of $\ddot x$ is increased
   2. The coefficient of $\ddot x$ is decreased
   3. The coefficient of $\dot x$ is increased a little
   4. The coefficient of $\dot x$ is increased a lot
   5. The coefficient of $\dot x$ is decreased
   6. The coefficient of $x$ is increased
   7. The coefficient of $x$ is decreased
   Verify your predictions by solving the equation and plotting the solution. Using a computer package is acceptable. Insight from problems of this type is very useful in designing feedback controllers in AME 30315.