Chapter Review Sheets for Boyce and DiPrima’s

1 Chapter Review Sheets for Boyce and diprima sElementary differential equations and Boundary Value Problems7th EditionChapter 1: IntroductionDefinitions: differential EquationMathematical ModelDirection (Slope) FieldEquilibrium SolutionInitial Condition, Initial Value Problem (IVP)General SolutionOrdinary differential Equation (ODE), Partial differential Equation (PDE)Order, Linear, Nonlinear, LinearizationImportant Skills:Derive differential equations that mathematically model simple problems. (Example 1, ; Alsosee )Construct a direction field for a first order ODE, and sketch approximate solutions.

2 (Example 2, )Graph the integral curves of a general solution (Example 2, )Know what an initial value problem is, and how to show a given function is a solution to one.(Example 2, )Know the difference between an ordinary differential equation and partial differential equation.( )Know how to classify differential equations as order, and linearity. ( & 19) Chapter 2: First Order differential EquationsDefinitions:First Order Ordinary differential EquationIntegrating Factor, Integral CurvesSeparableExistence and Uniqueness of SolutionsGeneral Solutions, Implicit SolutionsAutonomous, Logistic Growth, Equilibrium Solutions, Critical PointsExact ODET angent Line Method (Euler s Method)First Order Difference EquationTheorems:Theorem :Existence and uniqueness of solutions to linear first order ODE :Existence and uniqueness of solutions to first order ODE s),(ytfy= 00)(yty=.

3 Theorem :Existence and uniqueness of solutions to exact first order ODE :Restatement and elaboration of theorem Skills:Be able to determine if a first order differential equation is linear or nonlinear. Equation (3) onpage 30 gives the form for a linear the differential equation is linear, compute the integrating factor, and then the general solution.(Example 4, p. 36)Be able to graph integral curves for an ODE. (Example 4, p. 36)If it s nonlinear, is it separable? If it s separable, you will need to compute two different crucial to know integration of basic functions and integral methods from your calculus example, various substitutions, integration by parts, and partial fractions will all be utilized.

4 (Examples 2&3, p. 42 & 44)If the differential equation is not separable, is it exact? If so, solve it using the method in (Example 2, p. 92)If it isn t separable or exact, check for substitutions that would convert it into a linear equation, or anonlinear equation that is then separable. For example, exercises 27-31 in section , show howBernoulli equations can be transformed into linear how to obtain approximate solutions using Euler s method if an analytical solution cannot befound. (Example 2, )Understand the three steps in the process of mathematical modeling.

5 (Example 3 p. 54)Determine the existence and uniqueness of solutions to differential equations . (Example 2, p. 66)Know how to recognize autonomous equations , and utilize the direction field to represent solutionsto them. Be able to determine asymptotically stable, semi-stable, and unstable equilibriumsolutions. (Example 1, p. 80)Relevant Applications:Mixing Problems, Compound Interest, Motion in a Gravitational Field, Radioactive Carbon DatingChapter 3: Second Order Linear EquationsDefinitions:Homogeneous, NonhomogeneousCharacteristic EquationWronskianGeneral Solution, Fundamental Set of SolutionsLinear IndependenceParticular SolutionPeriod, Natural Frequency, Amplitude, PhaseOverdamped, Critically Damped, UnderdampedResonanceTransient Solution, Steady-State Solution or Forced ResponseTheorems:Theorem.

6 Existence and uniqueness of solutions to)()()(tgytqytpy=+ + , 00)(yty= 00)(yty = .Theorem :Principle of superposition. If 1y and 2y are solutions to 0)()(=+ + ytqytpy,so is 2211ycyc+ for any constants 1c and :Finding solutions to Eq. (2) an Eq. (3), using the Wronskian at the :Representing general solutions to second order linear homogeneous ODE sTheorem :Existence of a fundamental set of :Linear independence of functions and the :Abel s :Linear independence of solutions to )()()(tgytqytpy=+ + and :Relating differences in nonhomogeneous solutions to fundamental solutions.

7 (Used to prove the following theorem.)Theorem :General solutions to linear nonhomogeneous ODE :General solutions to linear nonhomogeneous ODE s. (Using variation ofparameters to determine the particular solution.)Important Skills:Be able to determine if a second order differential equation is linear or nonlinear, homogeneous ornonhomogeneous. (If it can be put into the form given by Equation (3) in page 130, it is linear.)Most of the Chapter deals with linear exceptions are two methods given in Section exercises 28-33, which show how tosolve second order differential equations missing the dependent variable, and Exercises 34-39,which shows how to solve equations missing the independent you recognize a homogeneous equation with constant coefficients, and derive thecharacteristic equation?

8 (Example 3, ) This equation will be quadratic, so know the quadraticformula, and the types of solutions one gets; real and distinct, repeated, and complex three cases will crucial to the types of solutions one gets to constant coefficienthomogeneous differential able to write down fundamental solution sets to homogeneous equations . This means findingtwo linearly independent solutions. You can use the Wronskian to show if two solutions arelinearly independent. (Example 3, page 141)Reduction of order is a way to take a know solution and produce a second linearly independentone.

9 Know it! (Example 3, )What are the fundamental solution sets for each of the three case of roots when solving constantcoefficient equations ? The summary is on (Example 3, ; Example 2, ; Example2 )Solutions to second order nonhomogeneous equations have two components. There is thehomogeneous solution, and particular, or nonhomogeneous solution. (Theorem )To find particular solutions you must know the method of undetermined coefficients, and variationof parameters. (Example 4, p. 173; Example 1, p. 180)Mechanical vibrations give excellent examples for utilizing all the techniques in the the difference between damped and undamped vibrations, forced and unforced the unforced case, if there is no dampening, the motion is sinusoidal.

10 Be able to determine thenatural spring frequency. (Example 2, ) If there is dampening, know the three different cases;underdamped, critically damped, and overdamped, depending on roots to the characteristicequation. If underdamped, know the quasi period. (Example 3, ) Know how to graphsolutions in the three different cases of the forced problem, the cases separate into damped or undamped. If undamped, there is thepossibility of resonance if the nonhomogeneous forcing term is sinusoidal with frequencyequivalent to the natural spring frequency.