n. an equation containing derivatives or differentials of a function. A partial differential equation (PDE) contains the partial derivatives of a function of more than one variable; otherwise the equation is an ordinary differential equation (ODE). First-order partial differential equations are reducible to systems of ODEs. Readily soluble first-order differential equations of the first degree include exact, separable, homogeneous, and linear differential equations. Readily soluble first-order differential equations of higher degrees are those which are algebraically soluble in the first derivative or either variable, and Clairaut's form. Second-order partial differential equations are fundamental to physics and include the wave equation, the heat equation, and Laplace's equation. The general quasi-linear equation of the second order is where A, B, and C can also be functions of u, u_x, or u_y; it is hyperbolic, parabolic, or elliptic depending upon whether B²- AC is negative, zero, or positive. The definitions can be extended to functions of more variables, but are of increasing complexity. There are methods for finding complete solutions of non-linear first-order partial differential equations if the equation explicitly contains either no dependent variable or no independent variable, or if it either has the form for some functions f and g, or has Clairaut's form. See Charpit's method, Jacobi's method. See also Lagrange's linear equation.