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Math · Advanced Math
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a) [30%) Given initial condition r()-1 and dr dt determine the time t at which r(t)too b) 30%) Consider the non-smooth differential equation: where f(y) f(y) and f(y) --I 0 + 1 for y > 0, for y 0, for y < 0 Determine the solution y(t) to this differential equation given initial condition y1 and y 1 at t O. Sketch the solution y(t) from time t 0 to t 2. c) [40%] The following differential equation arises naturally in physical problems involving oscillations: 2 dr2 where k is related to the wave-length A (period of a sinusoidal solution) by k 2T/A. The boundary conditions are y(0) 0 and y(L)0. First, is y() 0 a solution to the above problem? Then, find the eigenvalues kn and the eigenfunctions n(r) (also, called the non-trivial solutions). Sketch a few yn(r). Is this differential equation linear or nonlinear? Eulers identity: ecos0 isin . You may use whatever method to solve these questions

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