The following is a list of the major topics we have studied so far in this course. It is not presented in the same order as the textbook.

Topics:

Here are more detailed descriptions of some of these ideas and concepts.

(1) You should know what a differential equation (DE) is and what it means for a function to be a solution. You should understand the concepts of general vs. particular solutions, initial conditions, etc.

(2) You should understand how to interpret the vector field diagram for a DE of the form y'=f(x,y). (For example, you should be able to spot an "obvious" particular solution by looking at the vector field.) You should be familiar with the existence and uniqueness theorem for solutions to DE's and should be able to apply it to particular examples to say when solution curves exist and when they are unique. You should also be able to draw and use the phase diagram for an autonomous DE and understand the concepts of equilibrium solution, critical points (stable and unstable), etc.

(3) You should know how to solve DE's of the basic types which we studied in Chapter 1:

Separable equations: equations of the form y'=f(x)g(y). Solve by separating variables and integrating.

Linear equations: equations of the form y'+P(x)y=Q(x). Solve by (i) multiplying both sides of the equation by the integrating factor f(x)=e^(an antiderivative of P(x)), (ii) recognizing the left hand side of the new equation as the derivative of f(x)y, and (iii) integrating both sides of the equation and dividing by f(x) to get the general solution.

Exact equations: equations of the form M(x,y)dx+N(x,y)dy=0. Solve by finding a potential function F(x,y) so that dF(x,y)/dx=M(x,y) and dF(x,y)/dy=N(x,y).

Homogeneous equations: equations of the form y'=f(y/x). Solve by making the substitution y=xv, which will convert the equation into a linear equation.

More generally, you should know how to transform a DE into a new DE by making a substitution for the dependent variable.

Here are a few sample problems for you to try:

1. y' = x(1+y^2)

2. y' = y(1+x^2)

3. y'+4y = 2e^(-2x)

4. xy'+2y = x^(-1)+3x

5. (2x+3y) + (3x+2)y' = 0

6. (3x^2y^3+y^4) + (3x^3y^2+y^4+4xy^3)y' = 0

7. y' = (x-y)/(x+y)

8. y' = 2xy/(x^2-y^2)

9. y' = (4x+y)^2

(4) Generating and intepreting a mathematical model for a specific physical situation is one of the more difficult topics we considered. Remember the three-part procedure: (i) starting from a physical problem, derive a DE which serves as a mathematical model by a combination of physical principles, scientific laws, and common sense, (ii) solve the DE by one of the methods described above, and (iii) interpret your answer in the context of the specific problem under consideration.

Most of the types of problems we considered led to simple separable or linear equations. You should know how to model a problem in population growth, either by an exponential model or a logistic model. You should also be familiar with some of the other miscellaneous models we considered, i.e. heating/cooling problems, mixture and draining problems etc.