University of Notre Dame
Aerospace and Mechanical Engineering

AME 30314: Differential Equations, Vibrations and Controls I
Fall 2015
Homework 3

Problem 1
Determine whether each of the given functions, $x(t)$ is a solution to the associated differential equation.

1. $x(t) = \sin(t), \ddot{x} + x = 0$ .
2. $x(t) = \cos(2 t), \ddot{x} + x = 0$ .
3. $x(t) = 3 t + 2 / t , t^2 \ddot{x} + t \dot{x} - x = 0 , t \neq 0$ .
4. $x(t) = \cos(2 t) , x^2 \ddot{x} + \dot{x} / x = - 4 \cos^3 (2 t) - 2 \tan(2 t)$ .
5. $x(t) = t + 1/t + 5 t / 2 \ln(t) , t^2 \ddot{x} + t \dot{x} - x = t , t \neq 0$ .
   

Problem 2
Consider a uniform flexible rope of length $l$ and total mass $\rho l$ , as is illustrated on the right in Figure 1. Assume that the length of rope hanging off the end of the table is $x$ and that the coefficient of friction (both dynamic and static) between the rope and table is $\mu$ . Determine a differential equation with dependent variable $x$ and independent variable $t$ that describes the motion of the rope. What are the units for each term in the equation?

Figure: System for Problem 2

Problem 3
Based on Theorem D.1 in the textbook ¹, which of the following differential equations are guaranteed to have solutions that exist and are unique?

1. $\dot x = x$ where $x(0) = 0$ .
2. $\dot x^2 = x$ where $x(0) = 0$ .
3. $\dot x = \begin{cases}-1, & x \geq 0, \\ 1, & x < 0, \end{cases}$
   where $x(0) = 0$ .
4. $\dot x = \begin{cases}-1, & t \geq 5, \\ 1, & t < 5, \end{cases}$
   where $x(0) = 0$ .
5. $\dot x = x^2$ where $x(0) = 0$ .
6. $\dot x = x^{1/2}$ where $x(0) = 0$ .
7. $\dot x = -\left\vert x\right\vert$ where $x(0) = 0$ .
8. $\dot x = \sqrt{x^2 + 9}$ where $x(0) = 0$ .
   

Problem 4
Show that the set of functions $\left\{t^0, t^1, t^2, t^3, t^4, \ldots, t^n \right\}$ is linearly independent.

Problem 5
Show that

$$\left( 2 t^2 + 3 x \sin^2 t\right) \frac{d x}{d t} + 2x \left( t + x \sin t \cos t\right) = 0$$

is not exact, but when multiplied by $\mu(x,t) = x$ , it is exact. Find the solution. Leaving the solution in implicit form is fine.

Problem 6
One special case where it is possible to determine an integrating factor is when it only depends on $t$ . In such a case, the equation:

$$f(x,t)\frac{\partial \mu}{\partial t}(x,t)+ \mu(x,t)\frac{\partia... ...{\partial \mu}{\partial t}(x,t)- \mu(x,t)\frac{\partial g}{\partial t}(x,t)= 0$$

reduces to

$$f(x,t) \frac{d \mu}{d t}(t) + \mu(t)\frac{\partial f}{\partial t}(x,t) - \mu(t)\frac{\partial g}{\partial x}(x,t) =0$$

which gives

$$\frac{1}{\mu(t)}\frac{d \mu}{d t}(t) = \frac{\frac{\partial g}{\partial x}(x,t) - \frac{\partial f}{\partial t}(x,t)}{f(x,t)}.$$ (1)

1. Show that if we additionally have that the right-hand side of Equation () is only a function of $t$ , then $\mu(t)$ is given by
   $$\mu(t) = \exp \! \left(\int \frac{\frac{\partial g}{\partial x}(x) - \frac{\partial f}{\partial t}(x)}{f(x)} d x \right).$$

2. Show that $\left(e^t - \sin x \right) + \cos x(d x/d t) = 0$ is not exact, but that the above method to determine an integrating factor applies. Use that to find the solution.