(a) Verify that y1 = cos θ and y2 = sin θ satisfy the homogeneous linear second-order diff...

(a) Verify that y1 = cos θ and y2 = sin θ satisfy the homogeneous linear second-order differential equation  Since the set of solutions consisting of y1 and y2 is linearly independent, the general solution of the differential equation is y = c1 cos θ + c2 sin θ.

(b) Verify that y = eiθ, where i is the imaginary unit and θ is a real variable, also satisfies the differential equation given in part (a).

(c) Since y = eiθ is a solution of the differential equation, it must be obtainable from the general solution given in part (a); in other words, there must exist specific coefficients c1 and c2 such that eiθ = c1 cos θ + c2 sin θ. Verify from y = eiθ that y(0) = 1 and y′(0) = i. Use these two conditions to determine c1 and c2