Question

MATH4474: Introduction to Partial Differential Equations Revision Exercises chapter 2, 4-6 Q1. Consider the linear second order partial differential equation (i) (ii) (iii) Determine the class of this equation Find the characteristic coordinates Reduce the equation to canonical form 02. Consider the linear second order partial differential equation (a) [2 marks] determine the class of the equation (b) [2 marks] Find the characteristics of the equation. (c) 12 Marks] Sketch the characteristics in the (x, y) plane (d) [2 Marks] determine the characteristic coordinates (e) [9 marks] Reduce the equation to standard form (0) 13 marks] find its general solution.

Answer 1

2)

sketch characteristic equations:

y^5 4_xx + 2xy^3 4_xy + x^2_y u_yy + y 5 u_x + (y^3 - 2x^2 + xy^3)u_y = 0 comparing its with the standard eqn Rr + ss + TL + LL(x , y , z , p , z) = 0 R = y^5 s = 2xy^3 T = x^2y now s^2 - 4RT = 4x^2y^6 - 4x^2y^6 implies s^2 - 4RT = 0 implies given eqn is parabolic R lambda^2 + s lambda + T = 0 y^5 lambda^2 + 2xy^3 lambda + x^2y = 0 implies lambda = -2xy^3 plusminus squareroot 4x^2y^6 - 4x^2y^6/2y^5 lambda = -x/y^2 plusminus - lambda = -x/y^2, - x/y^2 dy/dx + lambda = 0 implies dy/dx - x/y^2 = 0 implies y^3/3 - x^2/2 = c_1 implies sigma(x , y) = y^3/3 - x^2/2 n(x , y) = y^3/3 + x^2/2 which are characteristic coordinates