Find a linear second-order constant-coefficient homogeneous

Question

MATH 54 QUIZ 10-

1. State Euler's identity. Solve for cosx and sinx in terms of e^ix and e^-ix.

2. Find a linear second-order constant-coefficient homogeneous ODE satisfied by y(x) = e^3xcos 2x.

3. Find the general solution to the differential equation y'' + 2y' - 3y = xe^x.

4. Solve the initial value problem y'' - y = sin x - e^2x, 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, . . . , y_m are linearly independent on (0, 1) if and only if they are linearly independent on (-1, 1).

(ii) Let y₁, . . . , y_n be functions on R. Then W[y₁, . . . , y_m](x) = 0 for all x ∈ R if and only if y₁, . . . , y_m 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.