In Math 1001, you learned about ordinary differential equations (ODEs):
equations which involve an unknown function and its derivatives. At the
time, you were exposed to very simple methods for solving ODEs, which
essentially amounted to using integration directly. In Math 2260, we'll
explore techniques for solving more complicated differential equations,
concentrating on ODEs which involve first and second derivatives. We'll
examine some applications of ODEs, as well. Although Math 2260 is chiefly
a computational course, we'll also give some consideration to the theory
behind the existence and uniqueness of solutions to these equations.
If you have any questions about the course, I can always be contacted at
[Your browser cannot view this email address; please turn on JavaScript] .
- This course has now concluded. Best of luck with your future
studies.
I do not make lecture notes available outside of class; if you miss a
lecture for any reason, I strongly encourage you to borrow the
corresponding notes from a classmate. However, as a guide, here is a brief
summary of what was covered in each lecture.
- May 13th: Ordinary, partial and other types of differential equations
(Section 1.1)
- May 15th: Order and linearity (Section 1.1); separable ODEs,
general and particular solutions, initial conditions and initial value
problems (Section 1.2)
- May 17th: Basic mathematical modelling, integral curves, direction
fields (Section 1.2)
- May 20th: First-order linear ODEs, the Method of Integrating Factors
(Section 1.3)
- May 22nd: The Method of Integrating Factors applied to IVPs
(Section 1.3); separable non-linear ODEs, “homogeneous”
ODEs (Section 1.4)
- May 24th: Making “homogeneous” ODEs separable, making
Bernoulli equations linear, non-linear ODEs which can be solved by several
methods (Section 1.4)
- May 27th: Exact equations, conditions for exactness, finding exact
solutions (Section 1.5)
- May 29th: Using integrating factors to make equations exact Section
1.5)
- May 31st: Conditions for existence and uniqueness of solutions to
linear and non-linear equations, intervals of definition (Section
1.6)
- June 3rd: Solutions of non-linear ODEs not arising from integration
(Section 1.6); mixing problems as an example of mathematicl
modelling (Section 1.7)
- June 5th: Parameters and long-term behaviour of solutions (Section
1.7); autonomous equations, fixed points, (asymptotically) stable and
unstable fixed points, the phase line, the carrying capacity of a
population (Section 1.8)
- June 7th: Modifying the logistic equation for a threshold effect,
bifurcations (Section 1.8)
- June 12th: A logistic equation with constant harvesting (Section
1.8); second-order linear equations with constant coefficients, the
characteristic equation (Section 2.1)
- June 17th: Second-order initial value problems (Section 2.1);
conditions for existence and uniqueness of solutions to second-order
linear equations (Section 2.2)
- June 19th: The Principle of Superposition, the Wronskian, fundamental
sets of solutions (Section 2.2); Euler's formula (Section
2.3)
- June 21st: Complex roots of the characteristic equation (Section
2.3); Repeated roots of the characteristic equation (Section
2.4)
- July 3rd: The method of reduction of order (Section 2.4)
- July 5th: Existence and uniqueness of solutions to nth-order linear
equations, fundamental sets of solutions for nth-order linear homogeneous
equations, general solutions of nth-order linear homogeneous equations
with constant coefficients (Section 2.5)
- July 8th: Initial value problems for nth-order linear homogeneous
equations (Section 2.5); solutions of second-order linear
non-homogeneous equations, the Method of Undetermined Coefficients
(Section 3.1)
- July 10th: Variations on the Method of Undetermined Coefficients
(Section 3.1)
- July 12th: Adjusting the form of the particular solution, applying the
Method of Undetermined Coefficients to nth-order equations (Section
3.1); the Method of Variation of Parameters (Section 3.2)
- July 15th: Applying the Method of Variation of Parameters (Section
3.2)
- July 17th: Variation of Parameters and equations with non-constant
coefficients, Variation of Parameters and nth-order equations (Section
3.2); modelling the motion of a mass on a spring, simple harmonic
motion (Section 3.3)
- July 22nd: Modelling the motion of a mass on a spring with damping and
forcing Section 3.3)
- July 24th: Integral transforms, the definition of the Laplace
transform, common Laplace transforms, the linearity of the Laplace
transform (Section 4.1)
- July 26th: The Shift Theorem (Section 4.1); Laplace transforms
of derivatives, using Laplace transforms to solve initial value problems,
inverse Laplace transforms (Section 4.2)
- July 29th: Inverse Laplace transforms and the Shift Theorem
(Section 4.2)
- July 31st: Predator-prey models, systems of coupled ODEs, using
Laplace transforms to solve systems (Section 4.2)
- August 2nd: The unit step function, Laplace transforms involving the
unit step function (Section 4.3)
- August 9th: Motion of a mass on a spring with a discontinuous forcing
function (Section 4.4)
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