13.5. EXERCISES 343

37. This and the following problems will present most of a differential equations course.Most of the explanations are given. You fill in any details needed. To begin with,consider the scalar initial value problem

y′ = ay, y(t0) = y0

When a is real, show the unique solution to this problem is y = y0ea(t−t0). Nextsuppose

y′ = (a+ ib)y, y(t0) = y0 (13.6)

where y(t) = u(t) + iv(t) . Show there exists a unique solution and it is given byy(t) =

y0ea(t−t0) (cosb(t− t0)+ isinb(t− t0))≡ e(a+ib)(t−t0)y0. (13.7)

Next show that for a real or complex there exists a unique solution to the initial valueproblem

y′ = ay+ f , y(t0) = y0

and it is given by

y(t) = ea(t−t0)y0 + eat∫ t

t0e−as f (s)ds.

Hint: For the first part write as y′− ay = 0 and multiply both sides by e−at . Thenexplain why you get

ddt

(e−aty(t)

)= 0, y(t0) = 0.

Now you finish the argument. To show uniqueness in the second part, suppose

y′ = (a+ ib)y, y(t0) = 0

and verify this requires y(t) = 0. To do this, note

y′ = (a− ib)y, y(t0) = 0

and that |y|2 (t0) = 0 and

ddt|y(t)|2 = y′ (t)y(t)+ y′ (t)y(t)

= (a+ ib)y(t)y(t)+(a− ib)y(t)y(t) = 2a |y(t)|2 .

Thus from the first part |y(t)|2 = 0e−2at = 0. Finally observe by a simple computationthat 13.6 is solved by 13.7. For the last part, write the equation as

y′−ay = f

and multiply both sides by e−at and then integrate from t0 to t using the initial con-dition.

38. Now consider A an n×n matrix. By Schur’s theorem there exists unitary Q such that

Q−1AQ = T