Explain the determine blue 5.5.2 Example BConsider the equationz(k +1, l + 1) + 2z(k, l) = k² + l +1,(5.170)or(E1 E2 + 2)z(k, l) = k² + l + 1.(5.171)The particular solution isZp(k, l) =1(k² + l +1)E, Eg + 2E1E2 + 211- (k² + l + 1)31+ 1/3(A1 + A2 + A¡A2)= 1/3[1–1/3(A1 + A2 + A¡A2)+ 1/9(A} + A3 + AA} + 2A¡A2 + 2A{A2+2A,A3)+ · · ·](k² +l+1)= 1/27(9k? – 6k + 9l + 5).(5.172)..The homogeneous equation(E1, E2 + 2)z½(k, l) = 0(5.173)has the solutionZn (k, l) = (-2)*f(e – k).(5.174)Therefore, the general solution of equation (5.170) isZh (k, l) = zh(k, l) + zp(k, l)(5.175)= (-2)* ƒ(l – k) + /27(9k² – 6k + 9l +5).

Given: The homogeneous eqution E1 , E2+2zhk, l=0 (5.173)has the solution zhk, l=-2k…

5.5.2 Еxample B Consider the equation z(k + 1, l + 1) + 2z(k, e) = k2 + l +1, (5.170) or (E, E2 + 2)z(k, l) = k² + l+ 1. (5.171) The particular solution is Zp(k, l) 1 (k² +l+1) E, E2 + 2 1 1 -(k² +l+1) 31+1/3(A1 + A2 + A¡A2) = 1/3[1-1/3(A1 + A2 + A¡A2) +1/o(△? + △2 + Δ우스2 + 2△1△2 + 2A{△2 +2Δ1Δ2)+.](k2 + l+ 1) = 1/27(9k? – 6k + 9l + 5). (5.172) The homogeneous equation (E1, E2 + 2)ž½(k, l) = 0 (5.173) has the solution zh(k, l) = (-2)* f (e – k). (5.174) Therefore, the general solution of equation (5.170) is Zh(k, l) = zh(k, e) + zp(k, l) (5.175) = (-2)* f(l – k) + /27(9k² – 6k + 9l + 5).

5.5.2 Еxample B Consider the equation z(k + 1, l + 1) + 2z(k, e) = k2 + l +1, (5.170) or (E, E2 + 2)z(k, l) = k² + l+ 1. (5.171) The particular solution is Zp(k, l) 1 (k² +l+1) E, E2 + 2 1 1 -(k² +l+1) 31+1/3(A1 + A2 + A¡A2) = 1/3[1-1/3(A1 + A2 + A¡A2) +1/o(△? + △2 + Δ우스2 + 2△1△2 + 2A{△2 +2Δ1Δ2)+.](k2 + l+ 1) = 1/27(9k? – 6k + 9l + 5). (5.172) The homogeneous equation (E1, E2 + 2)ž½(k, l) = 0 (5.173) has the solution zh(k, l) = (-2)* f (e – k). (5.174) Therefore, the general solution of equation (5.170) is Zh(k, l) = zh(k, e) + zp(k, l) (5.175) = (-2)* f(l – k) + /27(9k² – 6k + 9l + 5).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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