Commutative algebra in solving differential equations
نویسنده
چکیده
An overwhelming majority of systems dealt with in engineering applications is modelled as differential equations. Irrespective of whether it is a mechanical system, or an electrical system, or a thermal system, or a chemical system, or any hybrid of two or more of such systems, the behavior of such a system is often described using ordinary or partial differential equations (ODEs or PDEs, respectively). This apparent omnipresence of differential equations puts us up against the job of solving such equations, which at times can be quite formidable. One of the standard ways to tackle this problem is to write down the differential equations as first order equations. The ‘order’ of an ordinary differential equation is a natural number equal to the highest number of times that the unknown variable is differentiated in the equation. For example, the order of the equation d 3y dt3 + 6 2y dt2 + 11 dt + 6y = 0 is equal to 3. It had long been realized (more than two hundred years ago) that ODEs that have order equal to 1 are much easier to handle than higher order ones. Let us look at some of the benefits of first order representations.
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