Question

part b

iv. u(x,0)=0 8. (a) Find a general solution to the PDE Uay = 0. (Give your answer in terms of two arbitrary functions.) (b) Solve the nonlinear PDE (Au)? - 2uxxUyy = 0. Give your answer as general as possible, and note that this solution does not contain arbitrary functions (though it will have arbitrary coefficients). What is the key difference between this PDE and that in part (a)?

Answer 1

The given PDE is, Integrating both sides of 0 worty HX Ⓡy Again integrate both sides wirela, Us x , (y) + 02 (x) where di and I are arbitrary functions of t and x. (b) The non 'linear P.D.E. is, (Au)- 2 UxxUyy 50 0. Here .: (Uxxt Uyg) ² - 2uxx Myy = 2. t quant 2uxx > Unx + 4y = 0

Ils Auxialiary eq; IB, ( Put ux=m My = 1) m4+1=0 my -1 m = (-1) Y4 we find the fourth root of t. - element where n: 4 1. Now (2mtiori e n im: 0,1,2,3 ey =(Cos y to'sin ) (65 312 tosin 31) , Cos SR + Sims com (79 + c Sin 74) cos a tisins = 11 - 2 + 1 - 4 Com 39 + 'Sin 3g = cos (51-) + c'sin(1-4) - testosz Cos Supet e Sin Spe=60(8+1) + sin (948) Cod 77 +i Sin Ti - Cos (an –4) + i sin(an-4) 4 > m= (t + ) , ( 12 ) 2times 12 times

Here C.F. e roots are repeated are so we write roots as y = 0, lytm, x) + x +03 (y + max) tx 0₂(y+2, 2) du lyt mq x) . 7 + 0 Jo 4. (4+ (14.)x) + x 42 (% + ( 14.se) + ( 3 + (1-1) 1 × 4( 8 + ()*).