I derived four equations and put them into their matrix form. How can I solve for velocity with them?
$$ \begin{split} m_s \ddot{x}_s + R_{s1}\dot{x}_s + R_{2s}\left( \dot{x}_s - \dot{x}_m\right) + K_{s1}x_s + K_{s2} \left( x_s - x_m \right) &=& P_0 \\ m_m \ddot{x}_m + R_{m}\dot{x}_m - R_{2s}\left( \dot{x}_s - \dot{x}_m\right) + K_{m}x_m - K_{s2} \left( x_s - x_m \right)&=&0 \\ \dot{x}_s - v_s&=& 0 \\ \dot{x}_m - v_m &=& 0 \end{split} $$
$$ \left[ \begin{array}{cccc} 1 &0 & 0 &0 \\ 0 &1 & 0 & 0 \\ 0 & 0 & m_s &0 \\ 0 &0 & 0 & m_s \\ \end{array} \right] \left[ \begin{array}{c} \dot{x}_s \\ \dot{x}_m \\ \dot{v}_s \\ \dot{v}_m \end{array} \right] + \left[ \begin{array}{cccc} 0 &0 & - 1& 0 \\ 0& 0 &0 & -1 \\ K_{s1} + K_{s2} & -K_{s2} & R_{s1} + R_{s2} & -R_{s2} \\ -K_{s2} & K_{m} + K_{s2} & - R_{s2} & R_m + R_{s2} \end{array} \right] \left[ \begin{array}{c} {x}_s \\ {x}_m \\ {v}_s \\ {v}_m \end{array} \right] = \left[ \begin{array}{c} 0 \\ 0 \\ 1 \\0 \end{array} \right] \sin{\omega t} $$
Let \begin{align*} \mathbf{x}&:=\mathbf{x}(t)= \begin{bmatrix} x_s & x_m & v_s & v_m \end{bmatrix}^T,\\ A&:= \begin{bmatrix} 0 &0 & - 1& 0 \\ 0& 0 &0 & -1 \\ \frac{K_{s1} + K_{s2}}{m_s} & -\frac{K_{s2}}{m_s} & \frac{R_{s1} + R_{s2}}{m_s} & -\frac{R_{s2}}{m_s} \\ -\frac{K_{s2}}{m_s} & \frac{K_m + K_{s2}}{m_s} & - \frac{R_{s2}}{m_s} & \frac{R_m + R_{s2}}{m_s} \end{bmatrix},\\ \mathbf{c}&:=\begin{bmatrix} 0 & 0 & \frac1{m_s} &0 \end{bmatrix}^T \end{align*}
So your system is $$ \frac{\mathrm{d}}{\mathrm{d}t}\mathbf{x}+A\mathbf{x}=\sin(\omega t)\mathbf{c}. $$ Left multiply by $\exp(At)$ gives $$ \underbrace{\exp(At)\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t}+\exp(At)A\mathbf{x}}_{=\frac{\mathrm{d}}{\mathrm{d}t}[\exp(At)\mathbf{x}]}=\exp(At)\sin(\omega t)\mathbf{c} $$ So $$ \exp(At)\mathbf{x}(t)=\mathbf{x}(0)+\left(\int_0^t\exp(A\tau)\sin(\omega\tau)\,\mathrm{d}\tau\right)\mathbf{c} $$ or equivalently, $$ \mathbf{x}(t)=\exp(-At)\,\mathbf{x}(0)+\exp(-At)\left(\int_0^t\exp(A\tau)\sin(\omega\tau)\,\mathrm{d}\tau\right)\mathbf{c}. $$