Given differential equation is
\(\frac{d^2x}{dt^2}-4\frac{dx}{dt}+13x=0\)
It's auxiliary equation is
m2 - 4m + 13 = 0
⇒ m = \(\frac{4\pm\sqrt{16-52}}2\) = \(\frac{4\pm 6i}2\) = 2\(\pm\)3i
∴ x = e2t(C1 cos 3t + C2 sin 3t)
\(\frac{dx}{dt} \) = e2t(2C1 cos 3t + 2C2 sin 3t - 3C1 sin 3t + 3C2 cos 3t)
= e2t((2C1 + 3C2) cos 3t + (2C2 - 3C1) sin 3t)
Given that x(0) = 0
∴ C1 = 0
also given that \(\frac{dx}{dt}(0)=2\)
∴ 2C2 = 2
⇒ C2 = 2/3
∴ x = e2t(2/3 sin 3t)
⇒ 2/3 e2t sin 3t which is solution of given differential equation.