5.5.5 Example E Assume that c = 0 in equation (5.183) and continue to assume that F(k,l) is a solution to the homogeneous equation. Under these conditions, we have (aE1 + bE2)z(k, l) = F(k, l) (5.189) and (aE1 + bE2)F(k, l) = 0. (5.190) The solution to the last equation is F(k, €) = (-b/a)* f(e+ k), (5.191) where f is an arbitrary function of l + k. Examination of the left-hand side of equation (5.189) shows that it is of a form such that Laplace's method can be used to obtain a solution. If we let k +l = m = constant, Vk = z(k, l) = z(k, m – k), (5.192) then vk satisfies the first-order inhomogeneous equation avk+1 + bvk = (-b/a)*f(m), (5.193) where we have used the results of equations (5.191) and (5.192) to replace the right-hand side of equation (5.189). Note that f(m) is a constant. Solving for Vk gives k k - f(m)k (5.194) Vk = A | a where A is an arbitrary constant. Replacing m by l+k and A by an arbitrary function of l + k gives the complete solution to equation (5.189), under the assumption of equation (5.190),
5.5.5 Example E Assume that c = 0 in equation (5.183) and continue to assume that F(k,l) is a solution to the homogeneous equation. Under these conditions, we have (aE1 + bE2)z(k, l) = F(k, l) (5.189) and (aE1 + bE2)F(k, l) = 0. (5.190) The solution to the last equation is F(k, €) = (-b/a)* f(e+ k), (5.191) where f is an arbitrary function of l + k. Examination of the left-hand side of equation (5.189) shows that it is of a form such that Laplace's method can be used to obtain a solution. If we let k +l = m = constant, Vk = z(k, l) = z(k, m – k), (5.192) then vk satisfies the first-order inhomogeneous equation avk+1 + bvk = (-b/a)*f(m), (5.193) where we have used the results of equations (5.191) and (5.192) to replace the right-hand side of equation (5.189). Note that f(m) is a constant. Solving for Vk gives k k - f(m)k (5.194) Vk = A | a where A is an arbitrary constant. Replacing m by l+k and A by an arbitrary function of l + k gives the complete solution to equation (5.189), under the assumption of equation (5.190),
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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