(a) Show that it is impossible to find continuous functions a_1, a_0: R -> R so that the differentia ...

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(a) Show that it is impossible to find continuous functions a_1, a_0: R -> R so that the differential equation x''(t) + a_1(t) x'(t) + a_0(t) x(t) = 0 has the solution x(t) = t^2. (Hint: check whether the existence and uniqueness theorem applies.) (b) Consider a function f: R x C -> C and the complex initial value problem dz/dt = f(t, z), z(0) = z_0, for a complex-valued function z: R -> C and a complex number z_0. By considering real and imaginary parts of this equation, formulate conditions on f for the problem to have a unique solution. Show that z(t) = e^(it) and z(t) = cos t + i sin t both satisfy dz/dt = iz, z(0) = 1, and deduce Euler's formula e^(it) = cos t + i sin t.