The general form of a second order scalar linear PDE in two independent variables is , where  is the unknown function, the coefficients  are functions of the independent variables  and . The method of Laplace (not to be confused with integral transform methods of the same name) is a method which, when successful, will yield the general closed-form solution to such equations, depending upon two arbitrary functions of a single variable.

The method works by transforming the original PDE  into another second order scalar linear PDE  with the remarkable property that solutions of  can be found from solutions of  by differentiations and simple linear algebraic manipulations. In favorable circumstances solutions to equation  can be found and this then leads to solutions of the original equation.  If solutions to  cannot be found, then one may iterate the process to generate a sequence of equations  with the property that solutions to  can be constructed from solutions to  by differentiations and simple linear algebraic manipulations. The third optional argument in the calling sequence to Laplace specifies the number of iterations the procedure will calculate in attempting to arrive at a PDE  which can be integrated.  The default numberofiterations is 5.

The specific details of the method of Laplace are easiest to explain for equations of the type  (although this special form is not required for the procedure). For an equation of this form, define the Laplace invariants  and . One can show that if either  or  then the PDE can be integrated directly by linear ODE methods. If both  and  then the PDE can be easily transformed to the wave equation  and the solution thus found.

If  then one defines  and finds that: [1]  also satisfies an equation of the form ; and [2] the equation can be inverted to give . A similar transform can be defined if . See the examples for an explicit computation of these transforms.