(Solved): They are both part of  one question Rewrite the system of linear equations \[ \left[\begin{arr ...

They are both part of  one question

Rewrite the system of linear equations \[ \left[\begin{array}{l} x^{\prime} \\ y^{\prime} \end{array}\right]=\left[\begin{array}{cc} 2 & -3 \\ -4 & -1 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right] \] as a single second order differential equation for \( x \). \[ x^{\prime \prime}=x^{\prime}+x \] Tanks \( T_{1} \) and \( T_{2} \) contain 50 gallons and 25 gallons of salt solutions, respectively. A solution with 4 pounds of salt per gallon is pumped into \( T_{1} \) from an external source at \( 4 \mathrm{gal} / \mathrm{min} \), and a solution with 3 pounds of salt per gallon is pumped into \( T_{2} \) from an external source at \( 3 \mathrm{gal} / \mathrm{min} \). The solution from \( T_{1} \) is pumped into \( T_{2} \) at \( 2 \mathrm{gal} / \mathrm{min} \), and the solution from \( T_{2} \) is pumped into \( T_{1} \) at \( 1 \mathrm{gal} / \mathrm{min} \). Also, \( T_{1} \) is drained at \( 3 \mathrm{gal} / \mathrm{min} \) and \( T_{2} \) is drained at \( 4 \mathrm{gal} / \mathrm{min} \). Assume that both mixtures are well stirred. Let \( Q_{1}(t) \) and \( Q_{2}(t) \) be the number of pounds of salt in \( T_{1} \) and \( T_{2} \), respectively, at time \( t>0 \). Derive a system of differential equations for \( Q_{1} \) and \( Q_{2} \). (Use "Q_1" and "Q_2" as your variables.)