MATH 285: Worksheet 11 April 9, 2015 9.5/9.6 Main ideas The Method of Separation of Variables for Partial Differential Equations Two Important PDEs 1. Section 9.5: Heated Rods and the One-Dimensional Heat Equation Guiding principle: Differential Equation: Conditions: 2. Section 9.6: Vibrating Strings and the One-Dimensional Wave Equation Guiding principle: Differential Equation: Conditions: Section 9.7: The Two-Dimensional Heat and Wave Equations 1 The Method of Separation of Variables 1. 2. 3. 4. 5. 6. Recall 1. x00 + λx = 0, x(0) = x(L) = 0. Solution: 2. x00 + λx = 0, x0 (0) = x0 (L) = 0. Solution: Some Helpful Integrals Z 2 cos2 (2πx) dx = 1 • 0 Z • 2 2 cos (2πx) cos 0 2 • L Z L sin 0 nπx L nπx 2 dx = dx = 0 4 πn 0 n 6= 8 1 n=8 2 n even n odd Z nπx 2 L 0 n even • x cos dx = −4L n odd L 0 L π 2 n2 Z π 0 n even sin2 x sin (nx) dx = • 4 n odd n(4−n2 ) 0 2 Example: 5ut = uxx 0 < x < 10 t>0 ux (0, t) = ux (10, t) = 0 3 u(x, 0) = 4x Practice Problems 1. A is modelled by the differential equation below. Solve and explain what the conditions and any constants mean. 10ut = uxx 0<x<5 t>0 u(0, t) = u(5, t) = 0 u(x, 0) = 25 2. A is modelled by the differential equation below. Solve and explain what the conditions and any constants mean. ytt = 4yxx 0<x<π t>0 y(0, t) = y(π, t) = 0 y(x, 0) = sin x yt (x, 0) = 1 is modelled by the differential equation below. Solve and explain 3. A what the conditions and any constants mean. ytt = 25yxx 0<x<π t>0 y(0, t) = y(π, t) = 0 y(x, 0) = yt (x, 0) = sin2 x 4. A is modelled by the differential equation below. Solve and explain what the conditions and any constants mean. 3ut = uxx 0<x<2 t>0 ux (0, t) = ux (2, t) = 0 u(x, 0) = cos2 (2πx) 5. Find the Fourier sine and cosine series for f (t) = t(π − t), 0 < t < π and sketch the two graphs to which these series converge. 6. Find a formal Fourier series to x00 + 2x = t2 , x(0) = x(π) = 0. 7. Determine if resonance occurs in the system modelled below and find xsp and check for resonance. x00 + 4π 2 x = F (t) where F (t) is the odd function of period 2 with F (t) = 2t for 0 < t < 1 An Ending Thought: Perserverance is the hard work you do after you get tired of doing the hard work you already did. – Newt Gingrich 4
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