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3. y" + Ay = 0, 0

3. y" + Ay = 0, 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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Find Values of Lemda ( eigenvalues ) for which the given problem has a non trivial solution. Also determine the corresponding nontrivial solutions ( eigen function )

 

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+ Uyy = 0 Chapter 1. Preliminaries on Problems associated with Differential Equations 14 15 ch PDES %3D 4. y" + Ay = 0, 0<x < m/2, y'(0) = 0, y'(T/2) = 0. %3D pace coordi- DES, as far a associated both initial of each type %3D 5. y" = Ay = 0, 0<x < T, y(0) – y'(0) = 0, y(T) = 0. %3D %3D 6. y" – 2y + Ay = 0, 0<x < m, y(0) = 0, y(T) = 0. %3D 7. Discuss why the following PDES can not be solved by the method of sepa- ration of variables. space vari- tion), there (), where (as e.g. in che number J.r.t. space (b) Ugx + Uxy + Uyy = 0 %3D (a) uz + Uy = 1 8. Show that the PDE Upp + (1/r) Ur +(1/r²) uoo with u(r) = R(r) P(0) yields nditions is g²R" + rR' – XR = 0, P" + AP = 0 %3D where A is a constant. ch satisfies urier series continuous al solution. 9. Show that the PDE ,2 Utt + uz + u = a?u%z with u(x, t) = X (x)T(t) yields X" – \X = 0, T" + AT' + (1 – da²)T = 0 %3D where A is a constant. 10. Show that the PDE hey imply points are Uų = B(uxz + Uyy) with u(x, y, t) = X(x) Y (y) T(t) yields T' - BuT = 0, X" – XX = 0 %3D %3D and Y" + (A – µ)Y = 0 %3D where A, u are constants. .as a non- 11. Show that the PDE ons (eigen Upr + (1/r) ur + (1/r2) ug2 + Uzz = 0 with /r,0, z) = R(r) P(0) Z(z)
Transcribed Image Text:+ Uyy = 0 Chapter 1. Preliminaries on Problems associated with Differential Equations 14 15 ch PDES %3D 4. y" + Ay = 0, 0<x < m/2, y'(0) = 0, y'(T/2) = 0. %3D pace coordi- DES, as far a associated both initial of each type %3D 5. y" = Ay = 0, 0<x < T, y(0) – y'(0) = 0, y(T) = 0. %3D %3D 6. y" – 2y + Ay = 0, 0<x < m, y(0) = 0, y(T) = 0. %3D 7. Discuss why the following PDES can not be solved by the method of sepa- ration of variables. space vari- tion), there (), where (as e.g. in che number J.r.t. space (b) Ugx + Uxy + Uyy = 0 %3D (a) uz + Uy = 1 8. Show that the PDE Upp + (1/r) Ur +(1/r²) uoo with u(r) = R(r) P(0) yields nditions is g²R" + rR' – XR = 0, P" + AP = 0 %3D where A is a constant. ch satisfies urier series continuous al solution. 9. Show that the PDE ,2 Utt + uz + u = a?u%z with u(x, t) = X (x)T(t) yields X" – \X = 0, T" + AT' + (1 – da²)T = 0 %3D where A is a constant. 10. Show that the PDE hey imply points are Uų = B(uxz + Uyy) with u(x, y, t) = X(x) Y (y) T(t) yields T' - BuT = 0, X" – XX = 0 %3D %3D and Y" + (A – µ)Y = 0 %3D where A, u are constants. .as a non- 11. Show that the PDE ons (eigen Upr + (1/r) ur + (1/r2) ug2 + Uzz = 0 with /r,0, z) = R(r) P(0) Z(z)
Satisfies the given DE and the I.C's as well as B.Cs. If the expansion (i.e. Fourier serie appearing in the solution) converges to a continuous function with continue second order partial derivatives, then the formal solution is an actual solution Homogeneous and nonhomogeneous B.Cs. Boundary conditions of the form u(0, t) %3D %3D Tn = (1 'd)n 'In = are nonhomogeneous. When associated with the heat equation, they imply that the endpoints are kept at nonzero temperature. If the endpoints are insulated, then the B.Cs become homogeneous. 1.2.6 Exercises Find the values of (eigenvalues) for which the given problem has a non- trivial solution. Also determine the corresponding nontrivial solutions (eigen functions). 1. y" + Xy = 0, 0<x < T, y(0) = 0, y'(T) = 0. || %3D %3D 3. y" + Ay = 0, 0<x< 2m, y(0) = y(27), y'(0) = y'(2m).
Transcribed Image Text:Satisfies the given DE and the I.C's as well as B.Cs. If the expansion (i.e. Fourier serie appearing in the solution) converges to a continuous function with continue second order partial derivatives, then the formal solution is an actual solution Homogeneous and nonhomogeneous B.Cs. Boundary conditions of the form u(0, t) %3D %3D Tn = (1 'd)n 'In = are nonhomogeneous. When associated with the heat equation, they imply that the endpoints are kept at nonzero temperature. If the endpoints are insulated, then the B.Cs become homogeneous. 1.2.6 Exercises Find the values of (eigenvalues) for which the given problem has a non- trivial solution. Also determine the corresponding nontrivial solutions (eigen functions). 1. y" + Xy = 0, 0<x < T, y(0) = 0, y'(T) = 0. || %3D %3D 3. y" + Ay = 0, 0<x< 2m, y(0) = y(27), y'(0) = y'(2m).
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