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uagrange's method requires that pX² – µd + q²µ? = 0.

uagrange's method requires that pX² – µd + q²µ? = 0.

Advanced Engineering Mathematics
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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Explain the determine blue

5.3.3 Example C Let p+ q = 1 in the equation 2(k, l) = pz(k +1, l – 1) + qz(k – 1, l+1). (5.91) Lagrange's method requires that pA² – µd + q²µ² = 0. (5.92) The solutions, d1(H) and A2(u), of this equation are d1(14) = H, A2(4) = µq/p. (5.93) This gives the following two particular solutions: 21(k, l) = µk+l and z2(k, l) = (q/p)*µk+e. (5.94) 178 Difference Equations Summing these expressions and adding the results gives the general solution z(k, l) = g(k+ l)+ (q/p)*h(k+ l). (5.95) Note that the separation-of-variables method does not work for this equa- tion, since if z(k, l) = Ck De, the following result is obtained: CkDe = pCk+1De-1+qCk-1De+1, (5.96) and the necessary separation into expressions that depend only on k andl cannot be done.
Transcribed Image Text:5.3.3 Example C Let p+ q = 1 in the equation 2(k, l) = pz(k +1, l – 1) + qz(k – 1, l+1). (5.91) Lagrange's method requires that pA² – µd + q²µ² = 0. (5.92) The solutions, d1(H) and A2(u), of this equation are d1(14) = H, A2(4) = µq/p. (5.93) This gives the following two particular solutions: 21(k, l) = µk+l and z2(k, l) = (q/p)*µk+e. (5.94) 178 Difference Equations Summing these expressions and adding the results gives the general solution z(k, l) = g(k+ l)+ (q/p)*h(k+ l). (5.95) Note that the separation-of-variables method does not work for this equa- tion, since if z(k, l) = Ck De, the following result is obtained: CkDe = pCk+1De-1+qCk-1De+1, (5.96) and the necessary separation into expressions that depend only on k andl cannot be done.
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