classical methods

[logistic_guy]

Solve the following differential equation using classical methods and the given initial conditions.

\(\displaystyle \frac{d^2 x}{dt^2} + 2\frac{dx}{dt} + 2x = \sin 2t\)

\(\displaystyle x(0) = 2; \ \ \ \frac{dx}{dt}(0) = -3\)

 

[logistic_guy]

First we solve the homogeneous.

\(\displaystyle \frac{d^2 x}{dt^2} + 2\frac{dx}{dt} + 2x = 0\)

\(\displaystyle r = \frac{-2\pm\sqrt{2^2 - 4(1)(2)}}{2(1)} = -1\pm i\)

Then, the solution is:

\(\displaystyle x(t) = c_1 e^{-t}\cos t + c_2 e^{-t}\sin t\)