In certain physical models, the nonhomogeneous term, or forcing term, g(t) in the equation ay" + by' + cy = g(t) may not be continuous, but have a jump discontinuity. If this occurs, a reasonable solution can still beobtained using the following procedure. Consider the following initial value problem.240 if 0sts7T/6y" + 4y' + 40y = g(t); y(0) = 0, y'(0) = 0, where g(t) = .if t> 7x/6Complete parts (a) through (c) below.(a) Find a solution to the initial value problem for 0 sts 71/6.The solution for 0sts 7n/6 is y(t) = - 6 e -21*cos 6t - 2 e - 2tsin 6t + 6(Type an equation.)(b) Find a general solution for t> 71/6.The general solution for t> 7x/6 is y(t) = C, e-2 cos 6t + C, e -2t sin 6t(Type an equation. Do not use d. D. e, E, i, or I as arbitrary constants since these letters already have defined meanings.)(c) Now choose the constants in the general solution from part (b) so that the solution from part (a) and the solution from part (b) agree, together with their first derivatives, at t= 71/6. This gives a piecewisecontinuously differentiable function that satisfies the differential equation except at t= 7x/6.if 0st<7x/6The solution to the differential equation is y(t) = .if t> 7x/6

Answered: Image /qna-images/answer/c219afb4-1f56-4b43-87e3-6aaebb72a534.jpg

In certain physical models, the nonhomogeneous term, or forcing term, g(t) in the equation ay" + by' + cy = g(t) may not be continuous, but have a jump discontinuity. If this occurs, a reasonable solution can still be obtained using the following procedure. Consider the following initial value problem. 240 if 0sts 7x/6 y" + 4y' + 40y = g(t); y(0) = 0, y'(0) = 0, where g(t) = . if t> 7x/6 Complete parts (a) through (c) below. (a) Find a solution to the initial value problem for 0sts7r/6. The solution for 0sts 7/6 is y(t) = - 6 e cos 6t - 2 e 21 sin 6t + 6 (Type an equation.) (b) Find a general solution for t> 7a /6. The general solution for t> 7n/6 is y(t) = C, e -21 cos 6t + C2 e -21 sin 6t. (Type an equation. Do not use d, D, e, E, i, or I as arbitrary constants since these letters already have defined meanings.) (c) Now choose the constants in the general solution from part (b) so that the solution from part (a) and the solution from part (b) agree, together with their first derivatives, at t= 7x/6. This gives a piecewise continuously differentiable function that satisfies the differential equation except at t= 7x/6. if 0st< 7x/6 The solution to the differential equation is y(t) = if t> 7x/6

In certain physical models, the nonhomogeneous term, or forcing term, g(t) in the equation ay" + by' + cy = g(t) may not be continuous, but have a jump discontinuity. If this occurs, a reasonable solution can still be obtained using the following procedure. Consider the following initial value problem. 240 if 0sts 7x/6 y" + 4y' + 40y = g(t); y(0) = 0, y'(0) = 0, where g(t) = . if t> 7x/6 Complete parts (a) through (c) below. (a) Find a solution to the initial value problem for 0sts7r/6. The solution for 0sts 7/6 is y(t) = - 6 e cos 6t - 2 e 21 sin 6t + 6 (Type an equation.) (b) Find a general solution for t> 7a /6. The general solution for t> 7n/6 is y(t) = C, e -21 cos 6t + C2 e -21 sin 6t. (Type an equation. Do not use d, D, e, E, i, or I as arbitrary constants since these letters already have defined meanings.) (c) Now choose the constants in the general solution from part (b) so that the solution from part (a) and the solution from part (b) agree, together with their first derivatives, at t= 7x/6. This gives a piecewise continuously differentiable function that satisfies the differential equation except at t= 7x/6. if 0st< 7x/6 The solution to the differential equation is y(t) = if t> 7x/6

Physics for Scientists and Engineers: Foundations and Connections

1st Edition

ISBN:9781133939146

Author:Katz, Debora M.

Publisher:Katz, Debora M.

Chapter39: Relativity

Section39.2: Special Case: Galilean Relativity

Problem 39.2CE: Suppose the primed and laboratory observers want to measure the length of a rod that rests on the...

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