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
Classical Dynamics of Particles and Systems
5th Edition
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Stephen T. Thornton, Jerry B. Marion
Chapter9: Dynamics Of A System Of Particles
Section: Chapter Questions
Problem 9.17P
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