Question

4. Show that [ Ꮽ 8 and derive a general solution to 20°g , y og , €02 * @zot Ꭷ2 = 0.

Answer 1

& the general solution of It's Syndblic form is 28+ DD' - D² = 0 put Dam, Dal ant mal=0 me 1, ty geneed slation is g = 4, (* + m, 4) + de (t the Ge dilt -x) + delt tbx).

Answer 2

Consider the p.die. W ym 27 nyerere you one you are not a Yat almit, y) = 2 b(ust, 421, clat, 4) 3-1. b-qac 21+ 8 a 970. © is of hyperbolic type. can be written an. | 2 x + 1 + 194 2 0 -2 For the characteristic curvey de ! 11., 1271,-Ź. KUOPIO => x-ta G. De atat -6. So, the characteristic curvep are. (2,4)= uit in Chit aw+26. zu 21, 2-1, 1129, +22. x + 1 ) 2 241 3 0 = 0 , 1 2 V 7.0 so we can use the inverve mapping theorem Now, "I = Yesut Yn n = q + Yn. Y 2 794 + Ynntary +2%. Yux = zz + Yan Yan tynn. & Yąg +2 Yen tynn. - ③. Itt a 12 (-1) +2y n ) + 2 (-7 sn +2% nn). a Ygs - 49 sn + 4ynn .- t = -(Ysg + Yen ) +2 (Jan + n) 2 Ygg & Yan +2ynn. -6.

Putting the valux of ③ , @ 16 in @ we get & (se + 2 Ysy Yon) + (-Y2s + Ygy +2 Ynn) (2 95 - 4 Yay +94 ny)=0 ...(1-12 - Ź Yeg +(2+ Ź+2) Yan +(1+1-2) Yun 2o. - A -yn (Proved). Mos! Now we sowe 6. => 0. -> Film) ayz Echay. 2 y = SE(n) an + G(S). -> y F(4) + 6(3). [ht Stinjan FCM)). f(") + 6(3). Where F&G are arb.fr 9(3,n) = y(x, t) - G (-1) + F(x+2+). - This to the general solution.