[ylny + e^(-xy)]dx + (1/y - xlny)dy = 0

please solve this differential equation or tell me the type.

Answer 1

[Roger Stafford]

You can attempt to find an explicit solution using 'dsolve'. See its documentation at:

 http://www.mathworks.com/help/symbolic/dsolve.html

Presumably you would use something like

 dsolve('Dy = (y*log(y)+exp(-x*y))/(x*log(y)-1/y)','x')

or if you want y to be the independent variable

 dsolve('Dx = (x*log(y)-1/y)/(y*log(y)+exp(-x*y))','y')

However, there is no guarantee that 'dsolve' would succeed in finding an explicit solution. If not, you would need to solve it numerically using specified initial conditions with one of the ode functions, such as ode45, ode23, ode113, ode15s, etc.

[Walter Roberson]

I have been trying to figure out if there is a hidden variable here, that y is really y(t) or the like? The recursive definition is giving me difficulty in formulating the problem correctly.

[Roger Stafford]

No, this is an ordinary differential equation of the form

 dy/dx = f(x,y)

which can also be written

 dx/dy = 1/f(x,y)

It is possible that another parameter, t, could be defined that would simplify matters, but if so I couldn't find one.

[Walter Roberson]

Ah, I see, yes, it can be written

diff(y(x), x) = (y*ln(y)+exp(-x*y))/(-1/y+x*ln(y))

I still don't know how to solve that symbolically, but I do see now that a third variable is not needed to express it.