This is by no means a complete list of what one needs to learn, but it does
indicate the topics that will receive more emphasis.
| Section | Objectives |
|---|---|
| 1.1 | know what the following are: differential equation, solution of a DE, initial condition, initial value problem |
| verify solutions of a DE | |
| find a particular solution of an IVP given a general solution | |
| 1.2 | solve IVP's of the form dy/dx = f(x), y(x_0) = y_0 |
| use basic integration to solve velocity and acceleration problems | |
| 1.3 | understand what existence and uniqueness of solutions mean, know examples of nonunique solutions |
| understand the statement of the Existence and Uniqueness theorem for equations of the form dy/dx = f(x,y), and know how to apply it to examples | |
| 1.4 | know the definition of a separable equation, know how to solve a separable equation by integration |
| 1.5 | know the definition of a first-order linear equation, know how to solve a first-order linear equation using integration |
| be familiar with mixture and concentration problems as examples of how linear equations arise in simple situations | |
| understand existence and uniqueness for solutions of linear equations | |
| 1.6 | know the definition of an exact equation, know how to solve exact equations by integration, and the criterion for recognizing exact equations |
| know the definition of a Bernoulli equation, know how to solve a Bernoulli equation using substitution | |
| know the definition of a homogeneous equation, know how to solve a homogeneous equation using substitution | |
| 2.1 | know statement of the logistic equation; given the general solution, be able to work out behavior of typical situations governed by the logistic model |
| have some familiarity with the doomsday-extinction model | |
| 2.3 | have some familiarity with velocity-acceleration models involving air resistance |
| 3.1 | know the general theory of second-order linear differential equations: definition, homogeneous and nonhomegeneous equations, Principle of Superposition, Existence and Uniqueness Theorem, linear dependence, the Wronskian, the general form of the solutions of the homogeneous and nonhomogeneous cases |
| 3.2 | know the general theory of linear differential equations: definition, homogeneous and nonhomegeneous equations, Principle of Superposition, Existence and Uniqueness Theorem, linear dependence, the Wronskian, the general form of the solutions of the homogeneous and nonhomogeneous cases |
| 3.3 | know how to solve homogeneous linear differential equations with constant coefficients, using the characteristic equation |
| 3.4 | familiarity with free undamped and damped motion models, and understand the qualitative behavior in the underdamped and overdamped cases |
| know how to put a linear combination of sine and cosine functions into phase-angle form | |
| 3.5 | know the method of undertermined coefficients for solving a nonhomogeneous linear equation with constant coefficients, be able to write down the trial solution for any function of the form p(x) e^{rx} cos(kx) or p(x) e^{rx} sin(kx) |
| know how to use variation of parameters to find a particular solution of a nonhomogeneous linear equation, given the solutions of the associated homogeneous equation | |
| 3.8 | know definition of a boundary value problem, know definition of eigenvalue and eigenfunction |
| know how to find the eigenvalues and associated eigenfunctions for basic examples of boundary value problems | |
| 4.1 | know how to convert an nth-order DE to a system of first-order DE's |
| know how to solve simple systems by integration | |
| understand the phase-plane portrait representation of the solution of a system of two unknown functions | |
| know the E & U theorem for systems of linear DE's | |
| 4.2 | know the definition of a linear operator, and examples showing that L_1 L_2 need not equal L_2 L_1 if the operators do not have constant coefficients |
| know how to solve systems of linear equations by using linear operators (the method of elimination) | |
| 7.1 | know definition of the Laplace transform, be very familiar with Laplace transforms of the common functions |
| know the definition of the Heaviside step function, and its Laplace transform | |
| know the meaning of the inverse Laplace transform, and how to calculate it for common examples such as rational functions | |
| 7.2 | be familiar with the formulas relating Laplace transform to differentiation and integration |
| know how to use the Laplace transform to solve initial value problems for single equations and systems of equations | |
| 7.3 | be familiar with the translation formula, and how to use it to find inverse Laplace transforms |
| 7.4 | know the definition of the convolution of two functions, be familiar with the product formula |
| be familiar with the formulas for the derivative and the integral of a Laplace transform | |
| 7.5 | be familiar with the Laplace transform of a periodic function |
| be familiar with the t-translation formula | |
| 8.1 | review the theory of Taylor series, and understand the radius of convergence of a power series |
| know the definition of analytic function, know the example f(x)=exp(-1/x^2) (and f(0)=0) of a nonanalytic function: its derivatives all exist at x=0 (and have the value 0 there), but f(x) is not equal to the value of its Taylor series at any nonzero x | |
| understand the power series method for solving ordinary DE's and be able to apply it to examples | |
| 8.2 | know the definition of ordinary point and singular point for a second-order ordinary differential equation, know the behavior of the radius of convergence for power series solutions near ordinary points |
| be able to apply the power series method near ordinary points of second-order equations |