Show that the integral surface of the PDE x(y 2 + f)f x – y(x 2 + f) = (x 2 – y 2 )f which passes through the curve xf =a 3 , y = 0 is given by the form f 3 (x 3 + y) 2 = a 9 (x – y) 3 , where a is a constant. Show that a local solution of the partial differential equation f t +af x =f 2 subject to f(x, 0) = cos(x), where a is a constant can be written as Find the characteristic curves and the general solution of the linear equation f x + e x f y + e z f z = (2x + e x )e f
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