Robert Carmichael has written: 'On the general theory of the integration of non-linear partial differential equations' -- subject(s): Differential equations, Partial, Partial Differential equations
The global solution of an ordinary differential equation (ODE) is a solution of which there are no extensions; i.e. you can't add a solution to the global solution to make it more general, the global solution is as general as it gets.
Lawrence F. Shampine has written: 'Fundamentals of numerical computing' -- subject(s): Numerical analysis, Data processing 'The variable order Runge-Kutta code RKSW and its performance' -- subject(s): Runge-Kutta formulas 'Variable order Runge-Kutta codes' -- subject(s): Runge-Kutta formulas 'Theory and practice of solving ordinary differential equations (ODEs)' -- subject(s): Differential equations, Numerical solutions 'Variable order Runge-Kutta codes' -- subject(s): Runge-Kutta formulas 'A user's view of solving stiff ordinary differential equations' -- subject(s): Differential equations, Numerical solutions, Stiff computation (Differential equations) 'Linear equations in general purpose codes for stiff OKEs' -- subject(s): Differential equations, Numerical solutions 'Evaluation of implicit formulas for the solution of ODEs' -- subject(s): Implicit functions, Differential equations 'The variable order Runge-Kutta code RKSW and its performance' -- subject(s): Runge-Kutta formulas 'The variable order Runge-Kutta code RKSW and its performance' -- subject(s): Runge-Kutta formulas
M. Francaviglia has written: 'Applications of infinite-dimensional differential geometry to general relativity' -- subject(s): Differential Geometry, Function spaces, General relativity (Physics) 'Elements of differential and Riemannian geometry' -- subject(s): Differential Geometry, Riemannian Geometry
The answer will depend on the nature of the differential equation.
The machine proposed by Charles Babbage to perform calculations for differential equations is called the "analytical engine." Although it was never completed during his lifetime, it was designed as a general-purpose mechanical computer that could perform various mathematical operations, including solving differential equations. The analytical engine laid the groundwork for modern computing concepts.
Riemann equations, particularly in the context of Riemann surfaces and Riemannian geometry, are used in various fields such as mathematics, physics, and engineering. They are essential in complex analysis, where they help in understanding multi-valued functions and their properties. In physics, Riemannian geometry plays a crucial role in general relativity, describing the curvature of space-time. Additionally, they find applications in optimization problems and in the study of differential equations.
These are the general math courses in an undergraduate program of Mechanical Engineering. Actually, these are also the math courses required in ANY undergraduate Engineering curriculum: Algebra Trigonometry Analytic Geometry Differential Calculus Integral Calculus Mutivariable Calculus Differential Equations
the curvature of space time is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.
The book I used in college, and still use when needed, is A First Course in Differential Equations, by Dennis Zill. It's very clearly written with tons of problems and examples.The book Mathematics From the Birth of Numbers, by Jan Gullberg, is a cool book in general and also has a short and sweet introduction to ordinary differential equations (ODEs) at the end. He derives the general theories of ODEs pretty much entirely through the use of applications.Gradshteyn and Ryzhik's Table of Integrals, Series, and Products, which is a must-own book for mathematicians and scientists anyways, also has a rather short, but surprisingly detailed section on ODEs toward the end. I wouldn't recommend this for a novice, but it's a great reference to have once you've become familiar with differential equations.Mathematical Methods in the Physical Sciences, by Mary Boas, is a classic text covering many topics, including ODEs and PDEs (partial differential equations). I'd get this book simply for the immense amount of very useful topics it introduces in all the fields of mathematics, including the calculus of variations, tensor analysis, and functional analysis.Eventually, you'll need or want to learn about PDEs, and the most intuitive and comprehensible book I've seen regarding them is Partial Differential Equations for Scientists and Engineers, by Stanley Farlow. It's almost (if such a thing can be said about a rigorous math book) entertaining.
exact differential equation, is a type of differential equation that can be solved directly with out the use of any other special techniques in the subject. A first order differential equation is called exact differential equation ,if it is the result of a simple differentiation. A exact differential equation the general form P(x,y) y'+Q(x,y)=0Differential equation is a mathematical equation. These equation have some fractions and variables with its derivatives.
These are the general math courses in an undergraduate program of Mechanical Engineering. Actually, these are also the math courses required in ANY undergraduate Engineering curriculum: Algebra Trigonometry Analytic Geometry Differential Calculus Integral Calculus Mutivariable Calculus Differential Equations