Answer to Question #153212 in Differential Equations for Abbasali

Question

(Y^2+yz+z^2)dx+(z^2+zx+x^2)dy+(x^2+xy+y^2)dz

Answer 1

(y²+yz+z²)dx+(z²+zx+x²)dy+(x²+xy+y²)dz = (y²+z²)dx+(z²+x²)dy+(x²+y²)dz +d(xyz) = (x²+y²+z²)(dx+dy+dz) - (x²dx+y²dy+z²dz) - d(xyz) = (x²+y²+z²)d(x+y+z) - 1/3 d(x³+y³+z³ -3xyz)

As this form is symmetric with respect to the variables x, y and z, we should use the basic symmetric functions s=x+y+z, u=xy+yz+zx, v = xyz as new variables. We have then

x²+y²+z²=(x+y+z)² - 2(xy+yz+zx) = s²-2u

x³+y³+z³ -3xyz = (x+y+z)³ - 3(x²y + xy² + yz² + y²z + z²x + zx²) - 9xyz = (x+y+z)³ - 3(x+y+z)(xy+yz+zx) = s³-3su

(x²+y²+z²)d(x+y+z) - 1/3 d(x³+y³+z³ -3xyz) = (s²-2u)ds - 1/3 d(s³-3su) = (s²-2u)ds - (s²-u)ds +s du = s du - u ds = s² d(u/s) = -u² d(s/u)

So, with the integrating factor s^-2 = (x+y+z)^-2 the original differential form becomes integrable, and general solution of the differential equation, determined by this differential form, in the domain U₁ = {x+y+z not equal to 0} has the form f(u/s) = f((xy+yz+zx)/(x+y+z)), where f() is an arbitrary differentiable function.

In the domain U₂ = {xy+yz+zx not equal to 0} general solution of the differential equation has the form g(s/u) = g((x+y+z)/(xy+yz+zx)), where g() is an arbitrary differentiable function. The integrating factor in this domain is u^-2 = (xy+yz+zx)^-2.