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Mathematics LibreTexts

2: Equations of First Order

For a given sufficiently regular function \(F\) the general equation of first order for the unknown function \()\) is

$$F(x,u,\nabla u)=0$$

in \(n\). The main tool for studying related problems is the theory of ordinary differential equations. This is quite different for systems of partial differential of first order.

The general linear partial differential equation of first order can be written as

$$\sum_{i=1}^na_i(x)u_{x_i}+c(x)u=f(x)$$

for given functions \(a_i,\ c\) and \(f\). The general quasilinear partial differential equation of first order is

$$\sum_{i=1}^na_i(x,u)u_{x_i}+c(x,u)=0.$$

Thumbnail: Sir William Rowan Hamilton. Sir William Rowan Hamilton was an Irish physicist, astronomer, andmathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques. His best known contribution to mathematical physics is the reformulation of Newtonian mechanics, now called Hamiltonian mechanics.

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