Example: Back into the complex root section we complete the claim that

y₁ (t ) = e^lt cos(µt)        and      y₂(t) = e^lt sin(µt)

Those were a basic set of solutions.  Prove that they actually are.

Solution

Thus, to prove this we will require to take find the Wronskian for these two solutions and show that this isn't zero.

= e^lt cos(µt)( le^lt sin(µt) + µ e^lt cos(µt)) - e^lt sin(µt)( le^lt cos(µt) - µ e^lt sin(µt))

= µ e^2lt cos²(µt) + µ e^2lt sin²(µt)

= µ e^2lt( cos²(µt) + sin²(µt))

= µ e^2lt

Here, the exponential will never be zero and µ ≠ 0 whether it were we wouldn't have complex roots and so W ≠ 0. Thus, these two solutions are actually a fundamental set of solutions and hence the general solution in this case is. As:

 y (t ) = c₁e^lt cos (mt ) + c₂e^ltsin (mt)