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one of the oldest and most widely used techniques for solving some types of partial differential equations. A partial differential equation is called linear if the unknown function and its derivatives have no exponent greater than one and there are no cross-terms-i.e., terms such as f f' or f'f'' in which the function or its derivatives appear more than once. An equation is called homogeneous if each term contains the function or one of its derivatives. For example, the equation f'+f 2=0 is homogeneous but not linear, f'+x2=0 is linear but not homogeneous, and fxx+fyy=0 is both homogeneous and linear