Equations of first order of partial differential equation
Equations of first order [ edit ] Main article: First-order partial differential equation Linear and nonlinear equations [ edit ] Linear equations [ edit ] A PDE is called linear if it is linear in the unknown and its derivatives. For example, for a function u of x and y , a second order linear PDE is of the form � 1 ( � , � ) � � � + � 2 ( � , � ) � � � + � 3 ( � , � ) � � � + � 4 ( � , � ) � � � + � 5 ( � , � ) � � + � 6 ( � , � ) � � + � 7 ( � , � ) � = � ( � , � ) where a i and f are functions of the independent variables only. (Often the mixed-partial derivatives u xy and u yx will be equated, but this is not required for the discussion of linearity.) If the a i are constants (independent of x and y ) then the PDE is called linear with constant coefficients . If f is zero everywhere then the linear PDE is homogeneous , otherwis...