Download Course Proposal: Differential Equations

Course Proposal: Differential Equations
This document gives a rough week-by-week schedule of the learning outcomes in
terms of the type of problem that will be covered in terms of learning outcomes.
1. Recognize the form of a function and apply the appropriate rule for
differentiation. Recognise a differential equation and verify a solution.
Understand the constraint of initial condition. Boundary conditions. Solving the
simplest of differential equations. Problems: Differentiate, for
example, f (x) = eaix , xk cos kx. Which of the following are differential equations?
y03 + y = 0, y + x2 = 0, y00 + 3y0 + 5y = x2 . For each solution, find the particular
solution which satisfies (a) x = 0, y = 0, (b) x = 0, y0 = 0.
2. Recognize some standard types of differential equation: variables separable,
integrating factor, homogeneous. Knowledge of techniques to solve these
equations. Problems: State the type (or types) and order of each of the
followings differential equations: y0 + xy = 0, y0 + ex y = x2 , x2 y00 + xy0 + y = 0. Solve
these equations.
3. Integrate first order differential forms, the geometrical interpretation of the
solutions as a collection of integral curves or surfaces. Problems: Find the
solutions of 3x2 ydx + (y2 + x3 )dy = 0, (and corresponding questions in (x, y, z)).
Locate the curve for which x = 2, y = 3 and sketch its form.
4. Construct sketched direction fields for autonomous/non-autonomous first order
ordinary differential equations. Problems: Find the direction fields for (a)
ẋ = y, ẏ = x2 , (b) ẋ = x + x3 , ẏ = −y. Hence sketch the phase portraits in each case
5. Understand and use the techniques for solving second order linear differential
equations. Problem: Solve the differential equations y00 + 3y0 + y = 0 in the form
of a sum of exponentials.
6. Ability to calculate eigenvalues and eigenvectors of matrices. Problems:
Find
1, 2
the eigenvalues and corresponding eigenvectors for the matrix A =
.
3, 4
7. Revision and mid-term test
8. Understanding of matrix calculations to obtain solutions. Problem: Write the
differential equation y00 + 6y0 + y = 0 in matrix form, and find the general
solutions. Find the particular solution which satisfies x = 1, y = 2.
9. Interpret solutions in terms of pictures of integral curves to form phase portrait.
Knowledge of different types of phase portrait that can occur from eigenvalue
calculations. Problem: Find the phase portraits of the linear systems for the
various coefficient matrices A = . . ..
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10. Recognize the stability and instability in a phase portrait. Understand the
relevance of stability/instability in a dynamical system e.g. competing
populations and mechanics. Problem: For each matrix A in item 9, state whether
the corresponding origin is asymptotically stable, neutrally stable or unstable.
11. Recognize nonlinearity in systems. Find the fixed point(s) of simple nonlinear
systems. Classify the type and stability of the fixed point using the linearization
theorem. Use the information to sketch the simplest of nonlinear systems.
Problem: Find the fixed points of the system ẋ = x + x2 , ẏ = −y. Classify the fixed
points using the linearization theorem.
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