Math 44 – Differential Equations
Wednesday, February 21, 2007
What’s on the exam?
We’ll have an in-class exam on Monday, February 26. You can use a calculator but no other
references. The exam will include a reference sheet that we’ll compile today and publish Friday.
The exam covers the text material in Chapters 1-2, EXCEPT Sections 1.7 and 2.5 and the extended
examples in 1.2 (mixing problems) and 2.4 (damped oscillators). The problems in the text
(including review problems) are good practice and also a good source for exam problems. Other
topics covered: substitutions (in single equations and systems, especially Bernoulli equations);
existence and uniqueness; qualitative conclusions from direction fields.
Good policies:
(a) When you describe a solution, make clear for what interval of t-values it is valid.
(b) If asked to find a general solution, find all possible solutions; this will usually
involve arbitrary constants.
If asked to find a single solution, DON’T include arbitrary constants. Pick a
value, any value.
(c) Show work. (Sometimes necessary for full credit, usually necessary for part credit.)
Basics:
Recognize different kinds of equations and systems…
First order vs. higher order
Linear vs. non-linear
Autonomous vs. not
For single first-order equations, separable vs. not
For linear equations and systems, homogeneous vs. not
Solve single first-order equations if they are…
Separable (Sec. 1.2)
Linear (Sec. 1.8 – 1.9 or handout)
Bernoulli equations (homework problems 8-9)
(“Solving” may just mean reducing to integrals, but you should also be able to do the
integrals when they are reasonable)
Convert a high-order equation or system into a first-order system by adding extra variables. (For
example, convert the second-order mass-on-a-spring equation, Sec. 2.1, to a two-variable,
first-order system by adding v = y’, or convert the two-body planetary-orbit equation from
second-order, three variables to first-order, six variables.)
Solve initial value problems (Y’=F(Y,t) with Y(t0)=Y0) by finding values of constants
Solve systems if they are…
Partially decomposable (Sec. 2.3) and reduce to equations of the above type
The mass-on-a-string system (which comes from Y’’+Y=0) (Sec. 2.1)
Use Euler’s method to find values of solutions to initial-value problems,
whether for single equations (Sec. 1.4) or systems (Sec. 2.4)
Construct direction fields for single equations, and trace solutions using
the direction field, for single equations (Sec. 1.3)
…even if the equations are only partly defined
…by hand OR by software
Recognize the special features of direction fields (Sec. 1.3)
…when y doesn’t appear on the right side of the equation, and
…when t doesn’t appear (autonomous equations)
Construct vector fields and/or direction fields for autonomous two-variable
systems (Sec. 2.1-2.2), and trace out solutions, even if the systems
are only roughly described
…recognize equilibrium points, and whether they are
sources, sinks, saddles, or just centers
…recognize periodic solutions (when possible) and
whether they are attractors or non-attractors
Beyond the basics
Solve non-linear equations by finding fortunate substitutions (other than z = y^k)
Understand the existence theorem: An initial value problem (even for a system)
has a “solution” through (Y0, t0) if the function f is continuous in any
open region containing (Y0, t0). BUT the “solution” might be valid
only on a small interval.
Understand the uniqueness theorem: Suppose that in some open region,
the derivatives of f with respect to y (or all the coordinates of F
with respect to all the coordinates of Y) are all continuous. Then if
any two solutions whose graphs stay in the region agree at a single
point, they agree everywhere.
(Stronger: Same conclusion occurs if, in the open region, the slopes
|F(Y2,t)-F(Y1,t)| / |Y2-Y1| are bounded.)
Have a sense of when solutions are likely to veer off to infinity for finite t.
Be aware that Euler’s method will never warn you of that.
Have a sense of when Euler’s method is giving good answers --- namely, if
the derivative y’(t) isn’t changing much from one step to the next.
Understand both exponential growth models and logistic-curve models for
population growth.
Understand predator-prey models of the sort described in Sec. 2.1ff, to the
extent they can be understood.
(end)