Example 1. solve the pde 4ux – buy = 0, together with the auxiliary condition that u(0, y) = y3. by (2) we have u(x, y) = f(-3x – 4y). this is the general solution of the pde. setting x = 0 yields the equation y3 = f(-4y). letting w = - 4y yields f(w) = -3/64. therefore, u(x, y) = (3x + 4y) /64. solutions can usually be checked much easier than they can be derived. we check this solution by simple differen- tiation: uy = 9(3x + 4y)/64 and uy = 12(3x + 4y)? /64 so that 4ux – 3y = 0. furthermore, u(0, y) = (3.0 + 4y)3/64 = y3. o