PARTIAL DIFFERENTIAL EQUATIONS Give the general solution for

PARTIAL DIFFERENTIAL EQUATIONS

Give the general solution for U_xx + U_xy - 2U_yy = 0

Solution

The PDE au_xx + bu_xy + cu_yy = 0 is elliptic if 4ac > b² , hyperbolic if 4ac < b² . Thus, above equation is hyperbolic since 4 · 1 ·(2) < (1)2 , .

To solve it we factor ² x + xy 2 ² y = (x + 2y)(x - y). Since (x + 2y)(2x - y) = 0, (x - y)(x + y) = 0, we conclude that the general solution of the PDE is u(x, y) = (2x - y) + (x + y) with , arbitrary functions. One can also proceed by changing variables to new coordinates , ;

recall that one would want (x/ y )² + (x/ y) 2 = 0, obtained by simply replacing each x-derivative in the PDE by a factor of x, and each y-derivative in the PDE by a factor of y. x /y solves the same equation. This gives that

(x /y) = (-1 + 1 + 8)/ 2 = 1,

(x/ y) = (-1 1 + 8)/ 2 = 2, so we can take = x + y, = 2x - y, which gives the same general solution (an arbitrary function of plus one of ) as above