Worksheet 14 Solutions

# Worksheet 14 Solutions - MA 2930 May 4 2011 Worksheet 14 1...

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MA 2930, May 4, 2011 Worksheet 14 1. Recall the superposition principle. Which of the following diﬀerential equa- tions does it apply to? (a) y 00 + x 2 y 0 + e x y = 0 (b) y 000 + y 0 + y 2 = 0 (c) y 00 + 5 y = sin x (d) u xx = 5 u tt + xtu t The superposition principle says that if y 1 and y 2 are two solutions of a diﬀerential equation, then any linear combination c 1 y 1 + c 2 y 2 is also its solution. It applies to linear, homogeneous equations whether ODE or PDE. Therefore, it’s true of equations (a) and (d), but not of (b) and (c). 2. Which of these is an eigenvalue problem or would lead to one? (a) y 00 + 7 y = sin λx , y (0) = 0, y (2) = 0. (b) y 00 + λy 0 = sin x , y (0) = 0, y (2) = 0. (c) y 00 + 7 y = 0, y 0 (0) = λ , y (2) = 1 - λ . (d) u xx = 5 u tt + xtu t , u (0 ,t ) = 0, u (2 ,t ) = 0, u ( x, 0) = x , u t ( x, 0) = 0. If we deﬁne an eigenvalue problem as any boundary value problem (BVP) whose solutions depend upon a parameter, then (a), (b) and (c) are all eigen- value problems, the parameter being λ . There is no parameter apparent in the PDE (d), but upon separation of variables it gives rise to two ODEs which contain a parameter and constitute eigenvalue problems. 3. Find the eigenvalues and eigenfunctions of the following diﬀerential equation: y 00 + λy = 0 , y (0) - y 0 (0) = 0 , y (1) + y 0 (1) = 0 What is the general solution?

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Eigenfunctions are non-trivial solutions of the diﬀerential equation; the corresponding values of the parameter
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Worksheet 14 Solutions - MA 2930 May 4 2011 Worksheet 14 1...

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