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Differential Equations

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Moderate

2nd-Order Inhomogeneous Moderate 3

DIFFEQ-XXXN1C

Solve the initial value problem:

$$x''-x=\sin(t),\ x(0)=x_0,\ x'(0)=v_0$$

(that is, find an explicit formula for the solution in terms of the initial values $x_0$, $v_0$).

A

$x(t)=\sin(t)+x_0e^t+v_0e^{-t}$

B

$x(t)=-\cfrac{1}{2}\sin(t)+\left(\frac{2x_0+2v_0+1}{4}\right)e^t +\left(\cfrac{2x_0-2v_0-1}{4}\right)e^{-t}$

C

$x(t)=-2\sin(t)+(x_0+v_0)e^t+(x_0-v_0)e^{-t}$

D

$x(t)=-\cfrac{1}{2}\sin(t)+\left(\cfrac{x_0+v_0}{2}\right)e^t +\left(\cfrac{x_0-v_0}{2}\right)e^{-t}$