if this can be proved the relation ψ(x, y, z) − ƒ(z) = constant, will be the integral of the given differential equation. To prove this it is enough to show that, in virtue of ψ(x, y, z) = ƒ(z), the function ∂ψ/∂x − σZ can be expressed in terms of z only. Now in consequence of the originally assumed relations,
| ∂ψ | = μX, | ∂φ | = μY, | ∂φ | = μZ, |
| ∂x | ∂y | ∂z |
we have
| ∂ψ | / | ∂φ | = | σ | = | ∂ψ | / | ∂φ | , |
| ∂x | ∂x | μ | ∂y | ∂y |
and hence
| ∂ψ | ∂φ | − | ∂ψ | ∂φ | = 0; |
| ∂x | ∂y | ∂y | ∂x |
this shows that, as functions of x and y, ψ is a function of φ (see the note at the end of part i. of this article, on Jacobian determinants), so that we may write ψ = F(z, φ), from which
| σ | = | ∂F | ; then | ∂ψ | = | ∂F | + | ∂F | ∂φ | = | ∂F | + | σ | · μZ = | ∂F | + σZ or | ∂ψ | − σZ = | ∂F | ; |
| μ | ∂φ | ∂z | ∂z | ∂φ | ∂z | ∂z | μ | ∂z | ∂z | ∂z |
in virtue of ψ(x, y, z) = ƒ(z), and ψ = F(z, φ), the function φ can be written in terms of z only, thus ∂F/∂z can be written in terms of z only, and what we required to prove is proved.
Consider lastly a simple type of differential equation containing two independent variables, say x and y, and one dependent variable z, namely the equation