Solve a partial differentiation question

Question

The equation is expected to have a simultaneous equation down the solving line.

Comments

Where is the equation?

Answer 1

There is no specific system of equations, so I'll provide a simple example of simultaneous partial differentiation equations and solve the system.

EXAMPLE

(1) x∂z/∂x=y∂z/∂y

(2) 3z=y∂z/∂y

where z=h(x,y), a 2-variable function of x and y.

(1) contains three variables so it can't be solved without reference to (2). From this system we need to derive an equation which involves only two variables. Note that the right-hand side of each equation is the same so we can create another equation:

3z=x∂z/∂x, which only contains two variables, so it should be possible to solve it.

Separate the variables to make an integrable equation:

∂z/z=3∂x/x. If this had been dz/z=3dx/x it would integrate to ln(z)=3ln(x)+C where C is the integration constant. But because we actually have partial integration, C must be replaced by some function G(y), which is a one-variable function in y alone, so ∂G/∂x=0.

ln(z)=3ln(x)+G(y) is the result of partial integration.

G(y) can be replaced by ln(g(y)) and that gives us: ln(z)=3ln(x)+ln(g(y))=ln(x³g(y)), making z=x³g(y).

(2) can be integrated by simply separating the variables:

∂z/z=3∂y/y integrates ln(z)=3ln(y)+F(x) (similar to the logic used earlier) and ln(z)=ln(y³f(x)), where F(x)=ln(f(x)).

So we have z=y³f(x).

But z=y³f(x)=x³​g(y). Therefore, for equality, f(x) must be x³ and g(y) must be y³, and z=x³y³, which is the solution.

I hope this example helps.