Reference no: EM131075640
MATH 54 QUIZ 10-
1. State Euler's identity. Solve for cosx and sinx in terms of eix and e-ix.
2. Find a linear second-order constant-coefficient homogeneous ODE satisfied by y(x) = e3xcos 2x.
3. Find the general solution to the differential equation y'' + 2y' - 3y = xex.
4. Solve the initial value problem y'' - y = sin x - e2x, y(0) = 1, y'(0) = -1.
5. True or False? Justify your answer (if True, provide a proof; if False, provide a counterexample).
(i) Functions y1, . . . , ym are linearly independent on (0, 1) if and only if they are linearly independent on (-1, 1).
(ii) Let y1, . . . , yn be functions on R. Then W[y1, . . . , ym](x) = 0 for all x ∈ R if and only if y1, . . . , ym is linearly dependent.
(iii) Suppose y1 and y2 are solutions to the ODE y'' + 2y' - 6y = 0 on R. If y1(0) = y2(0) and y'1(0) = y'2(0), then y1(x) = y2(x) for all x ∈ R.
(iv) If y1 and y2 are solutions to y'' + y = sin x, then c1y1 + c2y2 is also a solution to y'' + y = sin x for any scalars c1, c2.
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