The Differential Equations

The Differential Equations

Synopsis

Each Problem Solver is an insightful and essential study and solution guide chock-full of clear, concise problem-solving gems. All your questions can be found in one convenient source from one of the most trusted names in reference solution guides. More useful, more practical, and more informative, these study aids are the best review books and textbook companions available. Nothing remotely as comprehensive or as helpful exists in their subject anywhere. Perfect for undergraduate and graduate studies. Here in this highly useful reference is the finest overview of differential equations currently available, with hundreds of differential equations problems that cover everything from integrating factors and Bernoulli's equation to variation of parameters and undetermined coefficients. Each problem is clearly solved with step-by-step detailed solutions. DETAILS- The PROBLEM SOLVERS are unique - the ultimate in study guides.- They are ideal for helping students cope with the toughest subjects.- They greatly simplify study and learning tasks.- They enable students to come to grips with difficult problems by showing them the way, step-by-step, toward solving problems. As a result, they save hours of frustration and time spent on groping for answers and understanding.- They cover material ranging from the elementary to the advanced in each subject.- They work exceptionally well with any text in its field.- PROBLEM SOLVERS are available in 41 subjects.- Each PROBLEM SOLVER is prepared by supremely knowledgeable experts.- Most are over 1000 pages.- PROBLEM SOLVERS are not meant to be read cover to cover. They offer whatever may be needed at a given time. An excellent index helps to locate specific problems rapidly. TABLE OF CONTENTSIntroductionUnits Conversion FactorsChapter 1: Classification of Differential EquationsChapter 2: Separable Differential EquationsVariable Transformation u = ax + byVariable Transformation y = vxChapter 3: Exact Differential EquationsDefinitions and ExamplesSolving Exact Differential EquationsMaking a Non-exact Differential Equation ExactChapter 4: Homogenous Differential EquationsIdentifying Homogenous Differential EquationsSolving Homogenous Differential Equations by Substitution and SeparationChapter 5: Integrating FactorsGeneral Theory of Integrating FactorsEquations of Form dy/dx + p(x)y = q(x)Grouping to Simplify SolutionsSolution Directly From M(x, y)dx + N(x, y)dy = 0Chapter 6: Method of GroupingChapter 7: Linear Differential EquationsIntegrating FactorsBernoulli's EquationChapter 8: Riccati's EquationChapter 9: Clairaut's EquationGeometrical Construction ProblemsChapter 10: Orthogonal TrajectoriesElimination of ConstantsOrthogonal TrajectoriesDifferential Equations Derived from Considerations of Analytical GeometryChapter 11: First Order Differential Equations: Applications IGravity and ProjectileHooke's Law, SpringsAngular MotionOver-hanging ChainChapter 12: First Order Differential Equations: Applications IIAbsorption of RadiationPopulation DynamicsRadioactive DecayTemperatureFlow from an OrificeMixing SolutionsChemical ReactionsEconomicsOne-Dimensional Neutron TransportSuspended CableChapter 13: The Wronskian and Linear IndependenceDetermining Linear Independence of a Set of FunctionsUsing the Wronskian in Solving Differential EquationsChapter 14: Second Order Homogenous Differential Equations with Constant CoefficientsRoots of Auxiliary Equations: RealRoots of Auxiliary: ComplexInitial ValueHigher Order Differential EquationsChapter 15: Method of Undetermined CoefficientsFirst Order Differential EquationsSecond Order Differential EquationsHigher Order Differential EquationsChapter 16: Variation of ParametersSolution of Second Order Constant Coefficient Differential EquationsSolution of Higher Order Constant Coefficient Differential EquationsSolution of Variable Coefficient Differential EquationsChapter 17: Reduction of OrderChapter 18: Differential OperatorsAlgebra of Differential OperatorsProperties of Differential OperatorsSimple SolutionsSolutions Using Exponential ShiftSolutions by Inverse MethodSolution of a System of Differential EquationsChapter 19: Change of VariablesEquation of Type (ax + by + c)dx + (dx + ey + f)dy = 0Substitutions for Euler Type Differential EquationsTrigonometric SubstitutionsOther Useful SubstitutionsChapter 20: Adjoint of a Differential EquationChapter 21: Applications of Second Order Differential EquationsHarmonic OscillatorSimple PendulumCoupled Oscillator and PendulumMotionBeam and CantileverHanging CableRotational MotionChemistryPopulation DynamicsCurve of PursuitChapter 22: Electrical CircuitsSimple CircuitsRL CircuitsRC CircuitsLC CircuitsComplex NetworksChapter 23: Power SeriesSome Simple Power SeriesSolutions May Be ExpandedFinding Power Series SolutionsPower Series Solutions for Initial Value ProblemsChapter 24: Power Series about an Ordinary PointInitial Value ProblemsSpecial EquationsTaylor Series Solution to Initial Value ProblemChapter 25: Power Series about a Singular PointSingular Points and Indicial EquationsFrobenius MethodModified Frobenius MethodIndicial Roots: EqualSpecial EquationsChapter 26: Laplace TransformsExponential OrderSimple FunctionsCombination of Simple FunctionsDefinite IntegralStep FunctionsPeriodic FunctionsChapter 27: Inverse Laplace TransformsPartial FractionsCompleting the SquareInfinite SeriesConvolutionChapter 28: Solving Initial Value Problems by Laplace TransformsSolutions of First Order Initial Value ProblemsSolutions of Second Order Initial Value ProblemsSolutions of Initial Value Problems Involving Step FunctionsSolutions of Third Order Initial Value ProblemsSolutions of Systems of Simultaneous EquationsChapter 29: Second Order Boundary Value ProblemsEigenfunctions and Eigenvalues of Boundary Value ProblemChapter 30: Sturm-Liouville ProblemsDefinitionsSome Simple SolutionsProperties of Sturm-Liouville EquationsOrthonormal Sets of FunctionsProperties of the EigenvaluesProperties of the EigenfunctionsEigenfunction Expansion of FunctionsChapter 31: Fourier SeriesProperties of the Fourier SeriesFourier Series ExpansionsSine and Cosine ExpansionsChapter 32: Bessel and Gamma FunctionsProperties of the Gamma FunctionSolutions to Bessel's EquationChapter 33: Systems of Ordinary Differential EquationsConverting Systems of Ordinary Differential EquationsSolutions of Ordinary Differential Equation SystemsMatrix MathematicsFinding Eigenvalues of a MatrixConverting Systems of Ordinary Differential Equations into Matrix FormCalculating the Exponential of a MatrixSolving Systems by Matrix MethodsChapter 34: Simultaneous Linear Differential EquationsDefinitionsSolutions of 2 x 2 SystemsChecking Solution and Linear Independence in Matrix FormSolution of 3 x 3 Homogenous SystemSolution of Non-homogenous SystemChapter 35: Method of PerturbationChapter 36: Non-Linear Differential EquationsReduction of OrderDependent Variable MissingIndependent Variable MissingDependent and Independent Variable MissingFactorizationCritical PointsLinear SystemsNon-Linear SystemsLiapunov Function AnalysisSecond Order EquationPerturbation SeriesChapter 37: Approximation TechniquesGraphical MethodsSuccessive ApproximationEuler's MethodModified Euler's MethodChapter 38: Partial Differential EquationsSolutions of General Partial Differential EquationsHeat EquationLaplace's EquationOne-Dimensional Wave EquationChapter 39: Calculus of VariationsIndex WHAT THIS BOOK IS FOR Students have generally found differential equations a difficult subject to understand and learn. Despite the publication of hundreds of textbooks in this field, each one intended to provide an improvement over previous textbooks, students of differential equations continue to remain perplexed as a result of numerous subject areas that must be remembered and correlated when solving problems. Various interpretations of differential equations terms also contribute to the difficulties of mastering the subject. In a study of differential equations, REA found the following basic reasons underlying the inherent difficulties of differential equations: No systematic rules of analysis were ever developed to follow in a step-by-step manner to solve typically encountered problems. This results from numerous different conditions and principles involved in a problem that leads to many possible different solution methods. To prescribe a set of rules for each of the possible variations would involve an enormous number of additional steps, making this task more burdensome than solving the problem directly due to the expectation of much trial and error. Current textbooks normally explain a given principle in a few pages written by a differential equations professional who has insight into the subject matter not shared by others. These explanations are often written in an abstract manner that causes confusion as to the principle's use and application. Explanations then are often not sufficiently detailed or extensive enough to make the reader aware of the wide range of applications and different aspects of the principle being studied. The numerous possible variations of principles and their applications are usually not discussed, and it is left to the reader to discover this while doing exercises. Accordingly, the average student is expected to rediscover that which has long been established and practiced, but not always published or adequately explained. The examples typically following the explanation of a topic are too few in number and too simple to enable the student to obtain a thorough grasp of the involved principles. The explanations do not provide sufficient basis to solve problems that may be assigned for homework or given on examinations. Poorly solved examples such as these can be presented in abbreviated form which leaves out much explanatory material between steps, and as a result requires the reader to figure out the missing information. This leaves the reader with an impression that the problems and even the subject are hard to learn - completely the opposite of what an example is supposed to do. Poor examples are often worded in a confusing or obscure way. They might not state the nature of the problem or they present a solution, which appears to have no direct relation to the problem. These problems usually offer an overly general discussion - never revealing how or what is to be solved. Many examples do not include accompanying diagrams or graphs, denying the reader the exposure necessary for drawing good diagrams and graphs. Such practice only strengthens understanding by simplifying and organizing differential equations processes. Students can learn the subject only by doing the exercises themselves and reviewing them in class, obtaining experience in applying the principles with their different ramifications. In doing the exercises by themselves, students find that they are required to devote considerable more time to differential equations than to other subjects, because they are uncertain with regard to the selection and application of the theorems and principles involved. It is also often necessary for students to discover those tricks not revealed in their texts (or review books) that make it possible to solve problems easily. Students must usually resort to methods of trial and error to discover these tricks, therefore finding out that they may sometimes spend several hours to solve a single problem. When reviewing the exercises in classrooms, instructors usually request students to take turns in writing solutions on the boards and explaining them to the class. Students often find it difficult to explain in a manner that holds the interest of the class, and enables the remaining students to follow the material written on the boards. The remaining students in the class are thus too occupied with copying the material off the boards to follow the professor's explanations. This book is intended to aid students in differential equations overcome the difficulties described by supplying detailed illustrations of the solution methods that are usually not apparent to students. Solution methods are illustrated by problems that have been selected from those most often assigned for class work and given on examinations. The problems are arranged in order of complexity to enable students to learn and understand a particular topic by reviewing the problems in sequence. The problems are illustrated with detailed, step-by-step explanations, to save the students large amounts of time that is often needed to fill in the gaps that are usually found between steps of illustrations in textbooks or review/outline books. The staff of REA considers differential equations a subject that is best learned by allowing students to view the methods of analysis and solution techniques. This learning approach is similar to that practiced in various scientific laboratories, particularly in the medical fields. In using this book, students may review and study the illustrated problems at their own pace; students are not limited to the time such problems receive in the classroom. When students want to look up a particular type of problem and solution, they can readily locate it in the book by referring to the index that has been extensively prepared. It is also possible to locate a particular type of problem by glancing at just the material within the boxed portions. Each problem is numbered and surrounded by a heavy black border for speedy identification.

Product details

If you enjoyed this product share it with others

Customer Reviews

• Be the first to review The Differential Equations

see all reviews