Question

Problem 1. Find the type, transform to normal form, and solve the following PDEs. (1) uxx – 16uyy = 0 - 2uxy + (2) Uxx Uyy = 0 (3) Uxx + 5uxy + 4uyy = 0 (4) Uxx – 6uxy + 9uyy = 0

Answer 1

Ans of D

Solutibu:- ou is - So, Here, which The gum equation U42 - 61 xy + quyy =0 S = -6, R=1, the coeffecients of uxy, lata respectively. (-6)2- 4x1x9 T-9 au and tlyy to Now, S²_URT 36 - 36 20 Sine, S? 4R7 0 Thus, we can say We in to that the given of parabolic type. Now, transform the given equation noumal forum as follows: As, the given equation is - 6uxy +94yy zo we first first find find char roots : and characteristic roots by Holving Rd² tsitt zo uan can be found out CS

Here, So, o Ral, T=9, Sc.6 26d+ 9 ? -3d-34+9=0 (1-3) -311-3) - (4-3) (1-3) = d=3,3 Now, consider + d = o du dy +-3 2-0 = de la dye-3 dx = dy -3dx Say = 6-3 de y = y tan - 3x + W +32 = 20 So one is of the characteristii uerte W = y + +3x crearly wwes must be be another let Since characteristic independent. So, haracteristic linearity anneindependent curve such that so are Scanned with

Now, AS w = y + 3* JW 3 gy Also, x ou OX 1 ay Now, ду. ду no ox + ou aur By Now, = DW du au + Ju an su 5304 + su 1 leming 9) 2w → 30 Now, Du² ou อ ЭХ 30 ow au 3 au 8u9 sh) (oring Ro q 2²u + 3 ore + zaru Ju mer CS Scanned, eith Cam + a²u 202 hner معان ماه QUOwl

Hence, au + 9ə*ч sus з ә4 wәU + 3.5*4 + у ә2- ə x2 anaw Now, OIL ди. Биз да + дц , а 27 ә8 ə o8 1) ди ә ( Using au au Әu (о "Я 9 ди- 1 Я Now, du әу? 4) әЯ о рц д Әч ( using Ф) ou ə uo 2 — әч (1o әу* aw² Now, alu әҹәЯ * (24) ( - )(%) CS Scanned with CamScanner (using ② and 6

Thus, за 2²u да - | | Fu *РЯ the equation 6 given equation 0 9 д% зә Put (10) in the fullsам . + за t +əq дио 2 18 əч цо- ОU дуу Әu2 дәU бә? CS Cams OuTu + 9ət д2

the So, 9 ใน Since, Because of continuity are awa Uaw above equation becomes :- 604 are -18 are bare Su2 Jugaw 2012 дц2 avaw +92u awe ㅋ CS Scarpe U2 Cants.com + +

Hence, due noumal the given equation :- tum of a²u 102 lou o Ans. we Now, Unr. 6 olay 20 Un > 1, will solve the given equation- +1uuu Put D, ly = D' D² - 6DD' +902 = 0 let D D Em So, m² 6m + 9 = 0 m²-3m-3m+9=0 mim-3) -3 (m-3)=0 (m-3)² m - 3,3 Hence, the solution of the given differentia equation is given by : u(x,y) = 0,14 +39) + * 2 (4+34)) Ans. CamScanner

For 2

Uax -2uxy tuyg - A Uxx + Buay + Cugy +D 4x +E vytfuza Mx = 1, y = -1 Now choosing another Chereefenste equation comparing with general form 9-7-7/-n is independent of ? To find the cononical Azt, 32-2, C=1, (0:1-F-=0) Jequation. Typeof PoE TA = APP ² + B8 x 8j + C qy? DE B2-4AC = (-2)2-4(1)(1) J(1)2+(-2) (131+1(172 1-2+ =0 [D=0 - PDE is Pasabduc 3=an8x+B (811 +2ynx) +2C&ply --261)(1) (1)+264.(-1) +101)) +24)(L)(-1) Clarecteristic equation = 2-2(-1+1)-2 : 2-200)-220 De = 4-4:0 ARO - Normal form The B B0 2A -2 2x1 2 dy dx = -1 Z = Anh + BMx ny + cnjr =(4)1432-2(1)(-1) + (-02 1 +2+134 c=4 = dy=f-da y=-x+C 19+x=(1) » There for party P=x+y Rail {ay=0 5 = Arx+ B&xy + clyy +D Extely Janco Exyzo lyg ac E = Anaat Bifry+Chay + D x + Eny Sezo Man ao nayo Mayzo 020, E20 P-620 GoGEOT D20, Eco CamScanner

Substituting these expressions into Augs+ Event Cung + B up + B Un + Fu =Ğ tougn +4 Unn +0.49 +0. un to u=0 = 4 unna Unno u = solving the above require migrating wire.n. un= f(9) Again integrating f(8).n+ g (9) Substitueting 2=x+4, n=x- u-f(x+y)(3-4) + g(1+y) 'U: (x-7f(x+y) + f(x+y) UCHY)=(x-3) fru+a+g(n+3) CS Scanned with CamScanner
Solution of first is