Mathematical Physics Vol 1

Chapter 7. Partial differential equations

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Coefficients next to (7.50) and (7.36) must be proportional, so we obtain ∂ u ∂ x P = ∂ u ∂ y Q = ∂ u ∂ z − c ′ R . We can determine c ′ from this system of equations, for example from ∂ u ∂ x P = ∂ u ∂ z − c ′ R . (7.51) Namely, it can be shown that if conditions v · rot v = 0 and rot v̸ = 0 ( v = P i + Q j + R k ) are satisfied, the equation (7.51) depends only on z , c ′ ( z ) and u ( x , y , z )= c ( z ) . A nonlinear first order differential equation (of two variables) is an equation of the form f ( x , y , z , p , q )= 0 , (7.52) where z = z ( x , y ) is the unknown function (of two independent variables x , y ), p = ∂ z ∂ x and q = ∂ z ∂ y . It is assumed that the function f , and all its partial derivatives, are continuous functions for all arguments, up to the required order. As we have already mentioned, there are three types of solutions: complete, singular and general. Complete and singular solutions Assume that the equation (7.52) can be obtained by elimination of two arbitrary constants (parameters), say a and b , from the function g ( x , y , z , a , b )= 0 . (7.53) Then the function g is called the complete solution (integral) of the partial equation (7.52). R Note. Geometrically, the complete solution defined in this way represents a two-parameter family of surfaces, which may or may not have an envelope. The envelope of these surfaces is also a solution of equation (7.52) and it is called the singular solution . The envelope (if it exists!) can be determined by eliminating constants a and b from the system of equations g = 0 , = 0 , = 0 . (7.54) (7.55) which satisfies the initial equation (7.52), then h ( x , y , z ) is called the singular solution (integral). If the function h ( x , y , z ) can be represented in the form of a product h ( x , y , z )= ξ ( x , y , z ) · η ( x , y , z ) , (7.56) where ξ = 0 satisfies the equation (7.52), while η = 0 does not, then ξ = 0 is a singular solution. A singular solution can be obtained from a partial equation by eliminating p and q fromthe system f = 0 , ∂ f ∂ p = 0 , ∂ f ∂ q = 0 . ∂ g ∂ a ∂ g ∂ b If by eliminating constants a and b from this system we obtain a function h ( x , y , z )= 0 ,

7.3.7 Nonlinear first order PDE. Lagrange-Charpit method