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`.

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