# Questions in Section Differential Equations: page 3

Given that y_1=e^t is a solution to the (d^3y)/(dt^3)+5(d^2y)/(dt^2)+5(dy)/(dt)-11y=0 differential equation. Find its general solution.

Use the Wronskian method to show the following solutions are linearly independent: y_1=e^x, y_2=xe^x, y_3=1.

Find a linear homogeneous constant-coefficient differential equation with the given differential solution: y(x)=c_1e^x+(c_2+c_3x+c_4x^2)e^(-x).

A swimming pool has a volume of 50 cubic meters. A mass C (in kg) of chlorine is dissolved in the pool water. Starting at a time t = 0, water containing a concentration of 0.1C/V chlorine is pumped into the swimming pool at a rate of 0.02 cubic meters per, and the water flows out at the same rate.

A modification of the logistic model is given by the model of Schaefer (dP)/(dt)=1/tau(1-P/K)P-EP.

The model, which was developed for the simulation of the development of fish populations, is equivalent to the logistic model for E = 0, where L=P(-oo)=0 is assumed for simplicity. The last term -EP takes into account (human) predation that reduces the rate of population growth. It is reasonable to consider this term to be proportional to P: the effect of predation will increase with the population density. The variables K, E<1/tau, and t are assumed to be non-negative and constant.

A tank contains 10 gallons of brine having 2 pounds of dissolved salt. Brine with 1.5 pounds of salt per gallon enters the tank at 3 gal/min, and the well-stirred mixture leaves at 4 gal/min.

1. Find the formula for the amount of salt in the tank at any time.
2. Find the concentration of the salt in the tank after 5 min (in pounds per gallon).

Solve the following first-order differential equation: (1/x-2/y)dx+x/y^2dy=0.

Solve the following first-order differential equation: (1/(1+x)+2xy)dx+(e^y+x^2)dy=0.

Solve the following first-order differential equation: (dy)/(dx)+y=3xy^-2.

Solve the following first-order differential equation: (dy)/(dx)-y/(x+3)=2x+6.