Example 8

Solve the complex initial value problem

z'+ (3 + i4)z = 0, z(0) = e¹⁺ⁱ².

The general solution is

z(t) = Ce^-(3+i4)t.

Substituting the initial value at

t = 0, e¹⁺ⁱ² = C.

The particular solution is then

z(t) = e¹⁺ⁱ² e^-(3+i4)t = e^1-3t+i(2-4t).

The solution may also be written in polar form using Euler's Formula,

z(t) = ^e1-3t cis(2 - 4t).