Overview of: Differential equation exam
The autonomous differential equation module exam for Math 1241 is based on part 5 from the Math 1241 thread.
Material for the exam
Solving linear equations (Introduction and solution to linear equations from Math 1241 thread).
- We can solve a linear autonomous differential equation such $\diff{x}{t}=ax$. The general solution is an exponential: $x(t)=Ce^{at}$.
- If $a \lt 0$, we have exponential decay. If $a \gt 0$, we have exponential growth.
- Given an initial condition $x(0)=x_0$, we can change the general solution with arbitrary constant $C$ to the specific solution $x(t)=x_0e^{at}$.
Solving a differential equation graphically (Graphical methods from Math 1241 thread).
- For the autonomous equation $\diff{u}{t} = g(u)$, determine where $u(t)$ is increasing, decreasing, or constant from a plot of $g(u)$. Use this information to sketch the solution $u(t)$. (Here we use $u$ for the state variable, but of course, the actual letter will be different for different problems.)
- One can summarize where $u(t)$ is increasing, decreasing, or constant using a phase-line diagram, with points for equilibria and arrows (a vector field) showing the direction of movement.
- Be able to go between a plot of $g$ versus $u$ (where $u$ is the horizontal axis) and plot of $u$ versus $t$ (where $u$ is the vertical axis). In the phase line, $u$ is also the horizontal axis.
Stability of equilibria (Stability of equilibria from Math 1241 thread).
- For the autonomous equation $\diff{u}{t} = g(u)$, the equilibria are the points where the rate of change is zero, i.e., the values of $u$ where $g(u)=0$.
- One can determine the equilibria analytically (solve $g(u)=0$) or graphically (find points where plot of $g(u)$ crosses the $u$-axis).
- From the graph of $g(u)$, or the vector field on the phase-line, determine the stability of equilibria.
- Use the stability theorem to determine stability of the equilibria without looking at the graph of $g(u)$. The stability theorem says for an equilibrium $u_e$ (with $g(u_e)=0$), the equilibrium is stable if $g'(u_e) \lt 0$ and unstable if $g'(u_e) \gt 0$.
Numerical solution (Numerical solution from Math 1241 thread).
- We can approximate the solution to an autonomous differential equation $\diff{v}{t} = f(v)$ using the Forward Euler algorithm.
- To use Forward Euler, divide time into interval of length $\Delta t$.
- If we know the value of the state variable $v$ at time $t = a$, we can take one Forward Euler step to estimate the value of $v$ at time $t=a+\Delta t$.
- Plug in the known value $v(a)$ into the right hand side of the differential equation, getting the number $f(v(a))$, which is the slope $\diff{v}{t}$ at time $t=a$.
- Assume that from $t=a$ to $t=a+\Delta t$, the slope doesn't change. In other words, approximate $v(t)$ by its linear approximation at $a$: $v(t) \approx L(t) = v(a) + f(v(a))(t-a)$.
- Calculate the value of $v$ at $t=a+\Delta t$ using the linear approximation.
- The resulting formula is that $v(a+\Delta t) \approx L(a+\Delta t) = v(a)+f(v(a))\Delta t.$
- Repeatedly apply Forward Euler steps to march forward in time. At each time step, calculate a new slope, march forward for a time step of $\Delta t$ with that slope (i.e., add $\Delta t$ times that slope to your previous value), and obtain an estimate for $v$ at the next time step.
Bifurcations (Bifurcations from Math 1241 thread).
- If a dynamical system depends on a parameter $\beta$, which we might write as $\diff{w}{t} = f(w,\beta)$, we can look for qualitative changes in the behavior of the dynamical system as we change $\beta$.
- If the stability or number of equilibria changes as we change $\beta$, we say we have a bifurcation.
- A bifurcation point is the value of $\beta$ where such a change occurs.
- Given a sequence of graphs of $f(w,\beta)$ or a sequence of phase lines of the dynamical system for different values of $\beta$, sketch a bifurcation diagram, which is a graph of equilibria on a plot of $w$ versus $\beta$.
- To draw a bifurcation diagram, draw the equilibria as curves, showing how the value of the equilibrium changes with $\beta$. Indicate which equilibria are stable and which are unstable. (Usually, we draw stable equilibria with solid lines and unstable equilibria with dashed lines.)
- Be able to interpret a bifurcation diagram. Given a bifurcation diagram, determine the phase line for different parameter values.
Study aids
Review problems
All questions that may appear in the applications of differentiation exam are available so that you can practice them. In both these problems and on the actual exam, the set of problems as well as values of numbers, variables, parameters, and other quantities are selected randomly. You will want to generate multiple versions of the problems to see the larger array of problems. Given the random nature, we cannot guarantee that you will actually see all the problems that will appear on the exam. But the more problems you work on, the greater the chance you will work on problems that will show up on the test.
The format of the exam questions may differ from the practice problems in that the exam problems may have been modified to remove any answer blanks and the instructions specific to entering answers in the right format for computer grading.
For all problems on this exam, you will be expected to show your work, and the grading will be based on the work that you show.
Free form questions
Review questions: Single autonomous differential equation problems contains short answer problems that reflect the format of questions as they would appear on the exam.
Computer-scored problems
Review questions: Simple linear differential equations practice, Review questions: Bifurcation of single differential equation practice, , and contain problems where the computer will the score your answer for you, so you can see how well you did. With the exception of the quizzes, these computer generated scores do not count toward your grade. When these questions appear on the exam, they may be modified to the short answer format.
Worksheets
The worksheets from the Math 1241 thread are also good review.
Exam rules
- You are allowed 50 minutes to take the exam.
- You are allowed to have one-half of one letter sized sheet of paper of notes. Double-sided is OK. No restrictions to what can be on the sheet of paper.
- You are allowed to have a calculator in the exam, including a graphing calculator.
- No textbook or electronic equipment (other than calculator) allowed.