a. Salt flows into tank A at a rate of
(<em>y</em>(<em>t</em>)/500 lb/gal) * (50 gal/min) = <em>y</em>(<em>t</em>)/10 lb/gal
and out at a rate of
(<em>x</em>(<em>t</em>)/500 lb/gal) * (50 gal/min) = <em>x</em>(<em>t</em>)/10 lb/gal
so the net rate of change of the amount of salt in tank A is
Similarly, you would find
b. We have
Notice that <em>x'</em> = -<em>y'</em>, so
Solve for <em>x</em>: the characteristic equation
has roots and , so
Again using the fact that <em>y'</em> = -<em>x'</em>, we then find
Given that <em>x</em>(0) = 0 and <em>y</em>(0) = 10, we find