Well if you're familiar with differential calculus the only new step you'd need to learn is how to use the Green's function method to invert the wave equation. It's your intuition that needs to be revised here. Second-order ordinary differential equations only have two linearly independent solutions. Second order partial differential equations can easily have an infinite number of linearly independent solutions. You see this with Maxwell's equations in vacuum for example: every plane wave propagating at c in any direction with any wavelength is a solution. That's an infinite number of plane wave solutions, all linearly independent.