Mixing Problems 1

Problem 1.

A tank contains $10$ L of water in which initially $y_0$ kg of salt is dissolved. Brine runs in $2$ L per minutes containing $30$% of salt per liter and runs out $2$ L per minutes. The mixture in the tank is kept uniform by stirring.
(a) Determine the amount of salt $y(t)$ in the tank at all times $t>0$.
     Show that $\displaystyle{\lim_{t\to\infty}}y(t)=y_{*}$ independently of $y_0$ where $y_{*}$ is the     equilibrium solution of the problem.
(b) If initially there is no salt in the tank, i.e., $y(0)=y_0=0$, determine $y(5)$.

Population Models 1

Problem 1.

The model is $$y'(t)=Ky(t)(M-y(t)),$$ where $K,M>0$ are constants. Determine the general solution if the initial value is $y(0)=y_0>0$.

Pendulum 1

Problem 1.

Describe the motion of the simple pendulum.

Heating and Cooling 1

Problem 1.

If the temperature of the bread is $120^oC$, of the air is $30^oC$ and $K=0.0366$, determine the temperature of the bread $60$ minutes later.

Electric Circuits 1

Fig.1. RC-circuit.

 

Problem 1.

Determine $I(t)$ for
(a) a constant electromotive force,
(b) a periodic electromotive force,
if $I(0):=I_0$ is given.