# MATH 215, Section 202

## Session 2019W, Term 2 (Jan - Apr 2020)

Last update: 2020-04-01 ( lecture 34, typos in lectures 32-1, 33-2, 30-1 corrected )

Instructor: Wayne Nagata
Email: nagata(at)math(dot)ubc(dot)ca
Office: Mathematics building, room 112
Office hours: M W F 1:00-1:50 pm, or appointment by email
Telephone: 604-822-2573 (no voice mail)

### Webwork and Homework Assignments

See the course Canvas page.

### Midterm Tests

Midterm tests will be in-class, at the regular lecture time and place. More details will be available on the course Canvas page approximately one week in advance.

### Final Examination

Link to the UBC final exam schedule. More details will be available on the course Canvas page in early April.
• Bring UBCcard (or other official photo I.D., if UBCcard is unavailable), it must be checked during the exam.
• Laplace transform table PROVIDED, trig identities (if required) provided.
• No books, notes, etc. allowed.
• No calculators, cell phones, etc. allowed. Numerical answers may be given as "calculator ready".
• Topics: all

### Learning Outcomes

This course is an introduction to ordinary differential equations (ODEs). By the end of the course, a student should be able to:
• write clear explanations of solutions to mathematical problems, showing logical steps and arguments
• draw or interpret a slope field for any first order ODE, draw approximate solution curves consistent with a slope field
• determine if a first order ODE is separable or linear or exact and if so, solve the ODE
• determine if a first order ODE is autonomous and if so, find all critical points (equilibrium solutions), draw the phase portrait, determine the stability of all critical points
• determine if there exists an integrating factor that makes a first order ODE into an exact equation and if so, find the integrating factor and solve the ODE
• solve an initial value problem (IVP) for a first order ODE
• set up, solve and analyze a mathematical model that involves a first order ODE
• use MATLAB to plot graphs of functions, or find roots of functions
• use MATLAB to plot slope fields or solution curves for a first order ODE
• use Euler's method to construct, by hand or with MATLAB, an approximation of the solution to an IVP for a first order ODE
• determine if a second order ODE is linear and if so, if it is homogeneous or nonhomogeneous
• determine if solutions of a homogeneous linear second order ODE are linearly independent or linearly dependent
• find the general solution of a homogeneous linear second order ODE, or solve an IVP for such an equation
• set up, solve and analyze a mathematical model that involves a constant coefficient homogeneous linear second order ODE (e.g. mass-spring system with or without damping, with zero external forcing)
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• find the general solution of a nonhomogeneous linear second order ODE (undetermined coefficients, variation of parameters), or solve an IVP for such an equation
• set up, solve and analyze a mathematical model that involves a constant coefficient nonhomogeneous linear second order ODE (e.g. mass-spring system with or without damping, with constant or sinusoidal external forcing)
• write a higher order ODE as an equivalent system of first order ODEs
• determine if a system of first order ODEs is linear and if so, if it is homogeneous or nonhomogeneous
• determine if solutions of a homogeneous linear system of first order ODEs are linearly independent or linearly dependent; form a fundamental matrix from linearly independent solutions; express the general solution of a homogeneous linear system in terms of a fundamental matrix
• find the general solution of a homogeneous linear system of first order ODEs, or solve an IVP for such a system
• for a 2-dimensional autonomous (constant coefficient) linear system of 1st order ODEs, classify the behaviour near the critical point (equilibrium solution) at the origin, draw the phase portrait
• find the general solution of a nonhomogeneous linear system of first order ODEs (variation of parameters), or solve an IVP for such a system
• determine if a 2-dimensional system of first order ODEs is autonomous and if so, find all critical points (equilibrium solutions), use linearization to classify (if possible) the local behaviour and find the stability (if possible) at each critical point; draw the global phase portrait (incorporating local phase portraits, near each critical point)
• =====================
• determine if a second order ODE is conservative and if so use this fact to draw the phase portrait for the associated system of first order ODEs
• use the definition to find the Laplace transform of a function f(t)
• find the inverse Laplace transform of a function F(s)
• use Laplace transforms to solve an IVP for a nonhomogeneous linear second order ODE, with a piecewise continuous and/or impulsive (delta function) nonhomogeneous (forcing) term
• to be continued (updated approximately weekly)

### Assessment of Learning Outcomes, Grading

See the course Canvas page.

### Lecture Notes

and corresponding sections in the textbook (links open in new tab)