Question

Please answer the following questions please show all work and write your answers neatly and thoroughly please. Also please make sure your answers are correct. Thank you I appreciate it!

2. Solve the PDE 4ux - 5y = 0, together with the auxiliary condition that u(0,y) = y.

Answer 1

The given PDE , 4 Ux – 5 uy = 0 -- • Lagrange's Solution of a linear PBE : det, a given Linear PDE is P ou + Q alue =R Then the simultaneous relenary differential equatón Corresponding to PDE is, According to the problem, p=4, Q = -5, R=0 . (a= constant). : dau= 0 7 u = c; do te doen 7 Sche + 4 dy=0 7 56 cm + Afdy = cz 7 5x+4y=22 The solution of the PDE 8 [c= constant 4, = + (en) → u = +(5* +4y). t 9 u (wy) - ¢ (5x+4y) ->Ü

Now, a=0 Now a= 0 » ulo,y) = 0 (48) 9 g = $(47) [** u(1) -8] 7 ore $ (47) +(2) |u(my) = (5x +44) which a from 16 the required solnton ,
Solution of a PDE using Lagrange's method. 4Ux - 5Uy = 0