(a) Consider the P.D.E. e**u, +e*u, - 3xu = e**u, in u =u(x, y). Show that...

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(a) Consider the P.D.E. e**u, +e*u, - 3xu = e**u, in u =u(x, y). Show that the equation is hyperbolic, and i) find the two families of associated characteristic curves; [8 marks] ii) illustrate each family with a sketch. (b) Given the surface f(x, y) = xsin(y /x), i) find ôf / ox , and confirm that at any point on the surface such that its x, y co-ordinates satisfy the simple relation y = x , then of / ôx = T; [8 marks] ii) evaluate ôf / oy at the point on the surface when x = t, y= n/2. (a) Consider the P.D.E. e**u, +e*u, - 3xu = e**u, in u =u(x, y). Show that the equation is hyperbolic, and i) find the two families of associated characteristic curves; [8 marks] ii) illustrate each family with a sketch. (b) Given the surface f(x, y) = xsin(y /x), i) find ôf / ox , and confirm that at any point on the surface such that its x, y co-ordinates satisfy the simple relation y = x , then of / ôx = T; [8 marks] ii) evaluate ôf / oy at the point on the surface when x = t, y= n/2.