Elements Of Partial Differential Equations By Ian Sneddon.pdf
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Elements Of Partial Differential Equations By Ian Sneddon.pdf
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In this chapter we shall discuss the properties of ordinary differential equations in more than two variables. Parts of the theory of these equations play important roles in the theory of partial differential equations, and it is essential that they should be understood thoroughly before the study of partial differential equations is begun. Collected in the first section are the basic concepts from solid geometry which are met with most frequently in the study of differential equations.
A third example of the occurrence of systems of differential equations of the kind (1) arises in analytical mechanics. In Hamiltonian form the equations of motion of a dynamical system of n degrees of freedom assume the forms
where P, Q, and R are given functions of x, y, and z. For that reason we study equations of this type now. In addition to their importance in theoretical investigations in physics they play an important role in the theory of differential equations, as will emerge later.
This curve refers to a particular choice of initial conditions; i.e., it is the curve which not only satisfies the pair of differential equations but also passes through the point (a,b,c). Now the numbers a, b, and c are arbitrary, so that the general solution of the given pair of equations will consist of the curves formed by the intersection of a one-parameter system of cylinders of which y = y(x) is a particular member with another one-parameter system of cylinders containing z = z(x) as a member. In other words, the general solution of a set of equations of the type (7) will be a two-parameter family of curves.
In some instances it is a comparatively simple matter to derive one of the sets of surfaces of the solution (2) but not so easy to derive the second set. When that occurs, it is possible to use the first solution in the following way: Suppose, for example, that we are trying to determine the integral curves of the set of differential equations (6) and that we have derived the set of surfaces (8) but cannot find the second set necessary for the complete solution. If we write
Geared toward students of applied rather than pure mathematics, this volume introduces elements of partial differential equations. Its focus is primarily upon finding solutions to particular equations rather than general theory.Topics include ordinary differential equations in more than two variables, partial differential equations of the first and second orders, Laplace's equation, the wave equation, and the diffusion equation. A helpful Appendix offers information on systems of surfaces, and solutions to the odd-numbered problems appear at the end of the book. Readers pursuing independent study will particularly appreciate the worked examples that appear throughout the text.