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324_pdfsam_math 54 differential equation solutions odd

324_pdfsam_math 54 differential equation solutions odd -...

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Chapter 5 5. In this problem, C = 0 . 01 F, L = 4 H, and R = 10 Ω. Hence, the equation governing the RLC circuit is 4 d 2 I dt 2 + 10 dI dt + 1 0 . 01 I = d dt ( E 0 cos γt ) = E 0 γ 4 sin γt . The frequency response curve M ( γ ) for an RLC curcuit is determined by M ( γ ) = 1 [(1 /C ) 2 ] 2 + R 2 γ 2 , which comes from the comparison Table 5.3 on page 290 of the text and equation (13) in Section 4.9. Therefore M ( γ ) = 1 [(1 / 0 . 01) 4 γ 2 ] 2 + (10) 2 γ 2 = 1 (100 4 γ 2 ) 2 + 100 γ 2 . The graph of this function is shown in Figure B.43 in the answers of the text. M ( γ ) has its maximal value at the point γ 0 = x 0 , where x 0 is the point where the quadratic function (100 4 x ) 2 + 100 x attains its minimum (the first coordinate of the vertex). We find that γ 0 = 175 8 and M ( γ 0 ) = 2 25 15 0 . 02 . 7. This spring system satisfies the differential equation 7 d 2 x dt 2 + 2 dx dt + 3 x = 10 cos 10
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