Solving separable partial differential equations

Partial differential equations have a reputation for being impossible to solve. And in many cases, this is true - they are extremely difficult to analytically solve for a general solution. However, when a partial differential equation is separable, it can be solved fairly straightforwardly, as a system of ordinary differential equations. Here is how to do so.

Consider the PDE:

$$ \frac{\partial f}{\partial x} = 3xy $$

We can write the PDE as:

$$ f(x, y) = g(x) h(y) $$

Therefore:

$$ \frac{\partial f}{\partial x} = g'(x) h(y) $$

So:

$$ g'(x) h(y) = 3xy $$

We can then separate:

$$ \frac{g'(x)}{3x} = \frac{h(y)}{y} $$

Now, an examination of this indicates that:

$$ \frac{g'(x)}{3x} = \frac{h(y)}{y} = \lambda $$

So we now have 1 ODE to solve, and one equation:

$$ g'(x) = 3x \lambda $$

$$ h(y) = \lambda y $$

The general solution of the ODE is:

$$ g(x) = \frac{3}{2} \lambda x^2 + C_1 $$

Given that $f(x, y) = g(x) h(y)$, we have the general solution of the PDE:

$$ f(x, y) = \frac{3}{2} \lambda^2 x^2 y + C_1 \lambda y $$