Mathematical Physics Vol 1

7.8 Examples

383

Problem 247 Find the general solution of equation

∂ f ∂ y

∂ f ∂ x

x

+ y

= 0 .

Solution To this PDE a system of equations is assigned of the form (see ( ?? ) onp. ?? ) d x x = d y y . It has been said that the solution is ψ i = C i , i = 1 ,..., n − 1 , where n is the number of independent variables, and ψ i are first integrals. In this case n = 2, and we thus have only one first integral ln y = ln x + ln c 1 ⇒ ln y = ln c 1 x ⇒ y = c 1 x ⇒ y x = c 1 , c 1 > 0 .

Thus, the general solution is of the form

f = f

y x

,

where f is an arbitrary differentiable function of the variable y / x .

Problem 248 Find the solution of the partial differential equation

∂ f ∂ x

∂ f ∂ y

∂ f ∂ z

( x + 1 )

+ y

+( z − 1 )

= 0 .

Solution To the given partial equation we assign the system of equations d x x + 1 = d y y = d z z − 1 , where n = 3, so there are n − 1 = 2 first integrals. First integrals are (from the first pair of equations) ln ( x + 1 )+ ln c 1 = ln y ⇒ c 1 = y x + 1 or C 1 = x + 1 y , that is (from the second pair of equations) ln y = ln ( z − 1 )+ ln c 2 ⇒ c 2 = y z − 1 or C 2 = z − 1 y , c 1 > 0 , c 2 > 0 .