Sample Solutions to Quiz 3 for MATH3270A October 31,2013 1.Find the general solution of the ODE: y (3) − y (2) + y 0 − Y = e−t sin t. Answer: The characteristic equation is: r3 − r2 + r − 1 = 0 The roots are r1 = 1,r2 = i,r3 = −i.The general solution of the homogeneous equation is: yh = c1 et + c2 cos t + c3 sin t A particular solution has the form yp = Ae−t sin t + Be−t cos t. Substitute it into the equation we get A = − 51 , B = 0. Thus the general solution is 1 y = c1 et + c2 cos t + c3 sin t − e−t cos t 5 2.Show that W [1, sin2 t, cos 2t] ≡ 0 for all t.Can you prove this without direct evaluation? Answer: Since cos 2t = cos2 t − sin2 t = 1 − 2 sin2 t,thus 1,sin2 t,cos2 t are linearly dependent,then Wronskian must be zero. 3.Find the Wronskian of a fundamental set of solutions to the ODE: ty (3) + 2y (2) − 5y = 0, t > 0 Answer: 1 2 By Abel’s formula: Z W (t) = c exp[− 2 dt] = c exp[−2 ln |t|] = ct−2 t
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