STUDY GUIDE: Ch. 1-2Boyce and DiPrima, Spring 2007This review is organized into four main areas: Theory, Analysis, Methods and Models.We’ve seen that a differential equation defines a family of functions (an initial value problem defines aspecific function). As such, ODEs provide a powerful tool for modeling.When we solve an ODE, we not only want to get an analytic solution, but we also want to understandgraphical analysis (direction fields, phase plots) and we want to be able to analyze the solution- Is the solutionunique? What is its behavior over the long term?, etc.1 Existence and Uniqueness Theorem1. Linear: y0+ p(t)y = g(t) at (t0, y0):If p, g are continuous on an interval I that contains t0, then there e xists a unique solution to the initialvalue problem and that solution is valid for all t in the interval I.2. General Case: y0= f(t, y), (t0, y0):(a) If f is continuous on a small rectangle containing (t0, y0), then there e xists a solution to the initialvalue problem.(b) If ∂f/∂y is continuous on that small rectangle containing (t0, y0), then that solution is unique.(c) We can only guarantee that the solution persists on a small interval about (t0, y0). To find the fullinterval, we need to actually solve the initial value problem.2 Analysis of Solutions1. Construct a direction field: Since y0= f(t, y), at each value of (t, y), we can compute the local slope,y0(t). Isoclines can be used to help: An isocline is determined by setting the derivative e qual to aconstant k and plotting the curve determined by: k = f(t, y).2. The Phase Diagram, and the Direction Field: Given that y0= f(y), we can plot y0vs. y. This gives usinformation that we can translate to the direction field, a plot of y vs t. This information is summarizedin the table below.In Phase Diagram: In Direction Field:y intercepts Equilibrium Solutions+ to − crossing Stable Equilibrium− to + crossing Unstable Equilibriumy0> 0 y increasingy0< 0 y decreasingy0and df/dy same sign y is concave upy0and df/dy mixed y is concave down3 MethodsThe primary way of solving first order ODEs is to first classify it by type, however, recall that a differen-tial equation may satisfy multiple classifications. For example, y0= ay + b is linear, separable, exact andautonomous.• Linear: y0+ p(t)y = g(t).Use the integrating factor: eRp(t) dtAlgebra: Recall that ea+b= eaeb, eln(A)= A1• Separation of variables: y0= f(y)g(t)Separate variables: (1/f(y)) dy = g(t) dt• Exact: M(x, y) + N(x, y)dydx, where Nx= My.Solution: Set fx(x, y) = M(x, y). Integrate w/r to x. Check that fy= N(x, y), and add a function of yif necessary.4 Models1. Construct an autonomous differential equation to model population growth in the standard model andwith an environmental carrying capacity. Be able to solve these models analytically (usually requirespartial fractions) and graphically.2. Be able to construct the differential equation corresponding to the tank mixing problem. Be able tosolve it (it will be a linear first order equation) and analyze it (using a phase diagram, if possible).3. Given Newton’s Law of cooling model, be able to find the coefficients in the model, solve it analytically,and analyze the behavior of the solutions.4. An object falling: mv0= mg − kv. For these problems, I will give you the constant(s) for g.5 Construct A Differential EquationIt is useful to construct your own differential equations so that you get a better feel for how they come about.This can be done through the modeling process, or directly. B e sure to try out the sample problems.6 Background MaterialLike algebra is to Calculus, Calculus is to ODEs. Here are some particular topics to remember:• Derivative formulas: The usual, but also the inverse trig formulas! It is possible to construct the formulayourself- For example:y = sin−1(x) ⇒ sin(y) = x ⇒ cos(y)dydx= 1 ⇒dydx=1cos(y)=1√1 − x2To write cos(y) in terms of x, draw a triangle that satisfies sin(y) = x, then determine cos(y) (which is√1 − x2) by the Pythagorean Theorem).• Integration Techniques: We’ve talked about integration by parts using a table, and partial fractiondecomposition.• Recall that f(x, y) = c determines y implicitly as a function of x. This idea was central to separableequations and exact equations.• The Fundamental Theorem of Calculus (Part I). If f (t) is continuous on the interval [a, b], then thefunction defined by:g(t) =Zt0f(x) dxexists, is continuous on [a, b] and differentiable on (a, b). In fact, g0(t) = f (t).This was used explicitly in a couple of places- In the Existence and Uniqueness Theorem for linear firstorder ODEs, and in Problem 14, p. 25 (Sect