Question

Tank 1 initially contains 50 gals of water with 10 oz of salt in it, while tank 2 initially contains 20 gals of water with 15 oz of salt in it. water containing 2 oz/gal of salt flows into tank 1 at a rate of 5 gal/min and the well-stirred mixture flows from tank 1 into tank 2 at the same rate of 5 gal/min. the solution in tank 2 flows out to the ground at a rate of 5 gal/min. if x1(t) and x2(t) represent the number of ounces of salt in tank 1 and tank 2, respectively, set up but do not solve an initial value problem describing this system.

Answer 1

`(\mathrm dx_1)/(\mathrm dt)=\frac{2\text{ oz}}{1\text{ gal}}\frac{5\text{ gal}}{1\text{ min}}-\frac{x_1(t)\text{ oz}}{50\text{ gal}}\frac{5\text{ gal}}{1\text{ min}}`

`(\mathrm dx_2)/(\mathrm dt)=\frac{x_1(t)\text{ oz}}{50\text{ gal}}\frac{5\text{ gal}}{1\text{ min}}-\frac{x_2(t)\text{ oz}}{20\text{ gal}}\frac{5\text{ gal}}{1\text{ min}}`


`\implies\begin{cases}(\mathrm dx_1)/(\mathrm dt)=10-\frac1{10}x_1\\\\(\mathrm dx_2)/(\mathrm dt)=\frac1{10}x_1-\frac14x_2\\\\x_1(0)=10\\\\x_2(0)=15\end{cases}`