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Use separation of variables to find a product solution to the following: d²u du -u, k>0 dr -U= dr² k

Use separation of variables to find a product solution to the following: d²u du -u, k>0 dr -U= dr² k

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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Question
Use separation of variables to find a product solution to the following: d²u du k- --u--, k>0 dx² dr u=e¹ ( A₁ e ² -ka 2 -ka 21 cosh ax+Be sinh ax u=e¹(A₂e² -kez 21 cos ax+B₂e" sin ax u=e¹(Azx+B₂) Ou=e"(Aphacosh ax +B phosinh x -kez 21 u=e =e-¹ (A₂e-ka₁²0 'cos ax + B₂e" sin ax x) u=e = e¯¹ (A₂x+B₂) Ou=e^¹ (A₁e² -ka² —ku sinh x cosh ax +Be -ka21 e¯¹(A₂¤¯ u=e cos ax+ B₂e sin ax u=e-¹(Azx+B₂) Ou=e^¹ (A₁e² -kez ²1 — ka-sinh (x cosh ax+B₁e" 21 u=e 2-¹ (A₂eka²10 cos ax+ Bekasin ax) u=e=¹' (Azx+B₂) ka Ou=el(1 phủ coshx+B pha sinh ax) u=e¹(A₂e-ka²1 c cos ax +Be-kasin ax) u=e¹(Azx+B₂) Ou=e¹(A₁e` -kez 21 -ka²1. cosh ax+Be sinh ax u=e¹(A₂eka²cos ax + B₁eka ²¹ sin ax) u=e¹(Azx+B₂) -ka21 -ka 2
Transcribed Image Text:Use separation of variables to find a product solution to the following: d²u du k- --u--, k>0 dx² dr u=e¹ ( A₁ e ² -ka 2 -ka 21 cosh ax+Be sinh ax u=e¹(A₂e² -kez 21 cos ax+B₂e" sin ax u=e¹(Azx+B₂) Ou=e"(Aphacosh ax +B phosinh x -kez 21 u=e =e-¹ (A₂e-ka₁²0 'cos ax + B₂e" sin ax x) u=e = e¯¹ (A₂x+B₂) Ou=e^¹ (A₁e² -ka² —ku sinh x cosh ax +Be -ka21 e¯¹(A₂¤¯ u=e cos ax+ B₂e sin ax u=e-¹(Azx+B₂) Ou=e^¹ (A₁e² -kez ²1 — ka-sinh (x cosh ax+B₁e" 21 u=e 2-¹ (A₂eka²10 cos ax+ Bekasin ax) u=e=¹' (Azx+B₂) ka Ou=el(1 phủ coshx+B pha sinh ax) u=e¹(A₂e-ka²1 c cos ax +Be-kasin ax) u=e¹(Azx+B₂) Ou=e¹(A₁e` -kez 21 -ka²1. cosh ax+Be sinh ax u=e¹(A₂eka²cos ax + B₁eka ²¹ sin ax) u=e¹(Azx+B₂) -ka21 -ka 2
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