Explain the determine blue 5.5.5 Example EAssume that c = 0 in equation (5.183) and continue to assume that F(k,l) isa solution to the homogeneous equation. Under these conditions, we have(aE1 + bE2)z(k, e) = F(k, l)(5.189)and(aE1 + bE2)F(k, l) = 0.(5.190)The solution to the last equation isF(k, €) = (-b/a)* f(l+ k),(5.191)where f is an arbitrary function of l + k.Examination of the left-hand side of equform such that Laplace's method can be used to obtain a solution. If we leton (5.189) shows that it is of ak + l = m = constant,Vk =z(k, l)= z(k, m – k),(5.192)then vk satisfies the first-order inhomogeneous equationavk+1 + bvk = (-6/a)kƒ(m),(5.193)where we have used the results of equations (5.191) and (5.192) to replace theright-hand side of equation (5.189). Note that f(m) is a constant. Solving forVk giveskkUk = A(5.194)awhere A is an arbitrary constant. Replacing m by l+k and A by an arbitraryfunction of l +k_gives the complete solution to equation (5.189), under theassumption of equation (5.190),kkkz (k, €) = (-2) g(e+ k) –f(l + k).(5.195)-aa

Given equation is aE1+bE2zk,l=Fk,l (i) Let us solve the equation using Laplace…

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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