Homework $6$ Problem $2$

A second order differential equation is given by

$\frac{d^2}{d{t}^2}y(t)+3\frac{d}{dt}y(t)+2y(t)=x(t)+\frac{d}{dt}x(t)$

Determine the roots of the charateristic equation where $s_1$ and $s_2$ satisfy $|s_1|<mi\; id="EditorElement\_374817">|</mi><msub\; id="EditorElement\_374826"><mrow><mi\; id="EditorElement\_374819">s</mi></mrow><mrow\; id="EditorElement\_374825"><mn\; id="EditorElement\_374822">2</mn></mrow></msub><mi\; id="EditorElement\_374828">|</mi>$

Determine the particular solution when the input $x(t)=sin(2t)u(t)$

Assuming all initial conditions are zero, determine the total solution $y(t).$