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. M 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 CONTENTS Introduction Units Conversion Factors Chapter 1: Classification of Differential Equations Chapter 2: Separable Differential Equations Variable Transformation u = ax + by Variable Transformation y = vx Chapter 3: Exact Differential Equations Definitions and Examples Solving Exact Differential Equations Making a Non-exact Differential Equation Exact Chapter 4: Homogenous Differential Equations Identifying Homogenous Differential Equations Solving Homogenous Differential Equations by Substitution and Separation Chapter 5: Integrating Factors General Theory of Integrating Factors Equations of Form dy/dx + p(x)y = q(x) Grouping to Simplify Solutions Solution Directly From M(x,y)dx + N(x,y)dy = 0 Chapter 6: Method of Grouping Chapter 7: Linear Differential Equations Integrating Factors Bernoulli's Equation Chapter 8: Riccati's Equation Chapter 9: Clairaut's Equation Geometrical Construction Problems Chapter 10: Orthogonal Trajectories Elimination of Constants Orthogonal Trajectories Differential Equations Derived from Considerations of Analytical Geometry Chapter 11: First Order Differential Equations: Applications I Gravity and Projectile Hooke's Law, Springs Angular Motion Over-hanging Chain Chapter 12: First Order Differential Equations: Applications II Absorption of Radiation Population Dynamics Radioactive Decay Temperature Flow from an Orifice Mixing Solutions Chemical Reactions Economics One-Dimensional Neutron Transport Suspended Cable Chapter 13: The Wronskian and Linear Independence Determining Linear Independence of a Set of Functions Using the Wronskian in Solving Differential Equations Chapter 14: Second Order Homogenous Differential Equations with Constant Coefficients Roots of Auxiliary Equations: Real Roots of Auxiliary: Complex Initial Value Higher Order Differential Equations Chapter 15: Method of Undetermined Coefficients First Order Differential Equations Second Order Differential Equations Higher Order Differential Equations Chapter 16: Variation of Parameters Solution of Second Order Constant Coefficient Differential Equations Solution of Higher Order Constant Coefficient Differential Equations Solution of Variable Coefficient Differential Equations Chapter 17: Reduction of Order Chapter 18: Differential Operators Algebra of Differential Operators Properties of Differential Operators Simple Solutions Solutions Using Exponential Shift Solutions by Inverse Method Solution of a System of Differential Equations Chapter 19: Change of Variables Equation of Type (ax + by + c)dx + (dx + ey + f)dy = 0 Substitutions for Euler Type Differential Equations Trigonometric Substitutions Other Useful Substitutions Chapter 20: Adjoint of a Differential Equation Chapter 21: Applications of Second Order Differential Equations Harmonic Oscillator Simple Pendulum Coupled Oscillator and Pendulum Motion Beam and Cantilever Hanging Cable Rotational Motion Chemistry Population Dynamics Curve of Pursuit Chapter 22: Electrical Circuits Simple Circuits RL Circuits RC Circuits LC Circuits Complex Networks Chapter 23: Power Series Some Simple Power Series Solutions May Be Expanded Finding Power Series Solutions Power Series Solutions for Initial Value Problems Chapter 24: Power Series about an Ordinary Point Initial Value Problems Special Equations Taylor Series Solution to Initial Value Problem Chapter 25: Power Series about a Singular Point Singular Points and Indicial Equations Frobenius Method Modified Frobenius Method Indicial Roots: Equal Special Equations Chapter 26: Laplace Transforms Exponential Order Simple Functions Combination of Simple Functions Definite Integral Step Functions Periodic Functions Chapter 27: Inverse Laplace Transforms Partial Fractions Completing the Square Infinite Series Convolution Chapter 28: Solving Initial Value Problems by Laplace Transforms Solutions of First Order Initial Value Problems Solutions of Second Order Initial Value Problems Solutions of Initial Value Problems Involving Step Functions Solutions of Third Order Initial Value Problems Solutions of Systems of Simultaneous Equations Chapter 29: Second Order Boundary Value Problems Eigenfunctions and Eigenvalues of Boundary Value Problem Chapter 30: Sturm-Liouville Problems Definitions Some Simple Solutions Properties of Sturm-Liouville Equations Orthonormal Sets of Functions Properties of the Eigenvalues Properties of the Eigenfunctions Eigenfunction Expansion of Functions Chapter 31: Fourier Series Properties of the Fourier Series Fourier Series Expansions Sine and Cosine Expansions Chapter 32: Bessel and Gamma Functions Properties of the Gamma Function Solutions to Bessel's Equation Chapter 33: Systems of Ordinary Differential Equations Converting Systems of Ordinary Differential Equations Solutions of Ordinary Differential Equation Systems Matrix Mathematics Finding Eigenvalues of a Matrix Converting Systems of Ordinary Differential Equations into Matrix Form Calculating the Exponential of a Matrix Solving Systems by Matrix Methods Chapter 34: Simultaneous Linear Differential Equations Definitions Solutions of 2 x 2 Systems Checking Solution and Linear Independence in Matrix Form Solution of 3 x 3 Homogenous System Solution of Non-homogenous System Chapter 35: Method of Perturbation Chapter 36: Non-Linear Differential Equations Reduction of Order Dependent Variable Missing Independent Variable Missing Dependent and Independent Variable Missing Factorization Critical Points Linear Systems Non-Linear Systems Liapunov Function Analysis Second Order Equation Perturbation Series Chapter 37: Approximation Techniques Graphical Methods Successive Approximation Euler's Method Modified Euler's Method Chapter 38: Partial Differential Equations Solutions of General Partial Differential Equations Heat Equation Laplace's Equation One-Dimensional Wave Equation Chapter 39: Calculus of Variations Index
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 aspect... ...Continua Nascondi