Tutorial Week 1
From UBCMATH WIKI
Worksheet Questions
- For each of the following differential equations, state whether it is an ODE or a PDE, state its order (if it’s a PDE, give the order in each independent variable) and whether it is linear or nonlinear.
- $t^3y'''+y'=\sin(t)$
- $\frac{\partial^2 y}{\partial x^2} + x^2 \frac{\partial y}{\partial t} = x^3$
- $e^x y+y \frac{d^2y}{dx^2} = x\cos(x)$
- Suppose that $y(t)=2+1/(1+t)+Ce^{3t}$ is the general solution to some differential equation. Consider the initial condition $y(0)=A$.
- For what values of $A$ does $\lim_{t\to+\infty}y(t) = \infty$?
- For what values of $A$ does $\lim_{t\to+\infty}y(t) = -\infty$?
- What, if any, are the values of $A$ for which this limit is finite?
- Sketch 3 members of this family of functions choosing values of A so that the behaviour of all types of solutions is demonstrated.
- Find solutions to the following ordinary differential equations. Recall that you can always check that you got the correct solution by plugging it back in to the given differential equation. Sketch the solution to the initial-value problem in part a. Show solution-checking process for the solution to the equation in part c.
- $x y'+3 y=x, \quad y(1)=0$
- $x y'+3 y=x^{-1} e^x$
- $y'=(1 + y^2)(3x^2-1)$ -- Remove this for next year (separable equations aren't covered until week 2)
Notes on Tutorial
- At the beginning of the section, explain the logistics of the tutorials.
- Working in groups is encouraged, but each student must complete and turn in their own worksheet at the end of the tutorial. (There's space to write answers between questions on the worksheet, encourage them to use it to make marking easier.)
- Worksheets will be marked by TAs and returned at the next tutorial.
- Students must come to the same tutorial each week. Verify by taking attendance that students are in the proper tutorial. If students are in the incorrect tutorial, make sure they know where to go next week, but allow them to stay.
- Give a brief overview of the problems and which WeBWork problems they are similar to.
- Question 1 is like WebWork problem 4.
- Question 2 is like pre-lecture WebWork problem 3.
- Question 3 is like WebWork problems 5 - 10.