I came across the following form of first order linear partial differential equation and I was wondering if there is a name for it?∂g∂x∂f∂y−∂g∂y∂f∂x=h(x,y),where g(x,y) as well as h(x,y) are given and f(x,y) is the unknown.

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faraidz3i · Accepted Answer

As Cameron Williams said, this is a first-order linear PDE, to which the method of characteristics can be applied. Your equation has a special form thanks to g; in fact, the left hand side of the equation is known as the Poisson bracket of f and g. A consequence of this is that the characteristic curves of the PDE are precisely the level sets g(x,y)=c.Let&#039;s see why. Let (x(t),y(t)) be any solution curve of the ODEx′(t)=−gy(x(t),y(t))y′(t)==gx(x(t),y(t))(I prefer using subscripts for derivatives.) This curve is a level curve of g, becauseddtg(x(t),y(t))=x′gx+y′gy=−gygx+gxgy=0Also, the derivative of unknown function f along this curve isddtf(x(t),y(t))=x′fx+y′fy=−gyfx+gxfy=h(x(t),y(t))Thus, the problem of solving the PDE splits into parametrizing level curves of g, and then integrating h along those curves.

I came across the following form of first order linear

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