Question

Solve the following Partial Differential Equation
yzp+2xq=xy

Answer 1

Answer:

For p = Z_x , q = Z_y the PDE is (yZ)Z_x + (2x)Z_y = xy . Apply the method

of characteristic writing the system

dx/ds = yZ

dy/ds = 2x

dZ/ds = xy.

From the second and third equations obtain ydy/ds =2xy = 2dZ/ds . Then

d(y^2)/ds = d(4Z)/ds and obtain the first integral

Z1 = y^2 - 4Z = K1.

From the first and third equations obtain xdx/ds = ZdZ/ds which

leads to the second integral

Z2 = x^2 - Z^2 = K2. The general integral of the equation is

F(Z1,Z2) = F(y^2 - 4Z , x^2 - Z^2) =0 with F arbitrary function

at least of class C1.