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1.6. FORCING TERMS: RESONANCES 30 25 25 20 20 15 시 10 5 0 α=0.02 α=0.01 α=0.00 115 0 0.2 0.4 0.6 0.8 1 w Figure 1.26: The ratio of output to input amplitudes w/VD as a function of the forcing frequency w for several choices of the damping factor a. 6. The Wronskian of two solutions Y₁(a) and Y2(x) to the homogeneous equation (1.6.34) is dy₁ dx W(x) = Y₁(x) x 2 (x) – Y2(x) (x). dY2 da Show that it satisfies the equation, - dW dx -(x) = -a W(x). Solve this equation with W(0) = Wo, and show that W(x) is never zero if Wo 0. What conclusion can you draw from this result?

1.6. FORCING TERMS: RESONANCES 30 25 25 20 20 15 시 10 5 0 α=0.02 α=0.01 α=0.00 115 0 0.2 0.4 0.6 0.8 1 w Figure 1.26: The ratio of output to input amplitudes w/VD as a function of the forcing frequency w for several choices of the damping factor a. 6. The Wronskian of two solutions Y₁(a) and Y2(x) to the homogeneous equation (1.6.34) is dy₁ dx W(x) = Y₁(x) x 2 (x) – Y2(x) (x). dY2 da Show that it satisfies the equation, - dW dx -(x) = -a W(x). Solve this equation with W(0) = Wo, and show that W(x) is never zero if Wo 0. What conclusion can you draw from this result?

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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1.6. FORCING TERMS: RESONANCES 30 25 25 20 20 15 시 10 5 0 α=0.02 α=0.01 α=0.00 115 0 0.2 0.4 0.6 0.8 1 w Figure 1.26: The ratio of output to input amplitudes w/VD as a function of the forcing frequency w for several choices of the damping factor a. 6. The Wronskian of two solutions Y₁(a) and Y2(x) to the homogeneous equation (1.6.34) is dy₁ dx W(x) = Y₁(x) x 2 (x) – Y2(x) (x). dY2 da Show that it satisfies the equation, - dW dx -(x) = -a W(x). Solve this equation with W(0) = Wo, and show that W(x) is never zero if Wo 0. What conclusion can you draw from this result?
Transcribed Image Text:1.6. FORCING TERMS: RESONANCES 30 25 25 20 20 15 시 10 5 0 α=0.02 α=0.01 α=0.00 115 0 0.2 0.4 0.6 0.8 1 w Figure 1.26: The ratio of output to input amplitudes w/VD as a function of the forcing frequency w for several choices of the damping factor a. 6. The Wronskian of two solutions Y₁(a) and Y2(x) to the homogeneous equation (1.6.34) is dy₁ dx W(x) = Y₁(x) x 2 (x) – Y2(x) (x). dY2 da Show that it satisfies the equation, - dW dx -(x) = -a W(x). Solve this equation with W(0) = Wo, and show that W(x) is never zero if Wo 0. What conclusion can you draw from this result?
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