91Ó°ÊÓ

Solve the given differential equation. $$y^{\prime \prime}-2 y^{\prime}=6 e^{3 x}$$

Use Euler's method with the specified step size to determine the solution to the given initial-value problem at the specified point. $$y^{\prime}=4 y-1, \quad y(0)=1, \quad h=0.05, \quad y(0.5)$$.

The number of bacteria in a culture grows at a rate that is proportional to the number present. Initially there were 10 bacteria in the culture. If the doubling time of the culture is 3 hours, find the number of bacteria that were present after 24 hours.

Determine the differential equation giving the slope of the tangent line at the point \((x, y)\) for the given family of curves. $$y=c e^{2 x}$$

A boy 2 meters tall shoots a toy rocket straight up from head level at 10 meters per second. Assume the acceleration of gravity is 9.8 meters/sec \(^{2}\). (a) What is the highest point above the ground reached by the rocket? (b) When does the rocket hit the ground?

A tank initially contains 600 L of solution in which there is dissolved 1500 grams of chemical. A solution containing \(5 \mathrm{g} / \mathrm{L}\) of the chemical flows into the tank at a rate of \(6 \mathrm{L} / \mathrm{min},\) and the well-stirred mixture flows out at a rate of 3 L/min. Determine the concentration of chemical in the tank after one hour.

The number of bacteria in a culture grows at a rate that is proportional to the number present. After 10 hours, there were 5000 bacteria present, and after 12 hours, there were 6000 bacteria present. Determine the initial size of the culture and the doubling time of the population.

A certain cell culture has a doubling time of 4 hours. Initially there were 2000 cells present. Assuming an exponential growth law, determine the time it takes for the culture to contain \(10^{6}\) cells.

In the logistic population model \((1.5 .3),\) if \(P\left(t_{1}\right)=P_{1}\) and \(P\left(2 t_{1}\right)=P_{2},\) then it can be shown (through some tedious algebra to derive by hand, although easy on a computer algebra system) that $$\begin{aligned}r &=\frac{1}{t_{1}} \ln \left[\frac{P_{2}\left(P_{1}-P_{0}\right)}{P_{0}\left(P_{2}-P_{1}\right)}\right] \\\c=& \frac{P_{1}\left[P_{1}\left(P_{0}+P_{2}\right)-2 P_{0} P_{2}\right]}{P_{1}^{2}-P_{0} P_{2}}\end{aligned}$$ These formulas will be used. The initial population in a small village is \(500 .\) After5 years, this has grown to \(800,\) while after 10 years the population is \(1000 .\) Using the logistic population model, determine the population after 15 years.

A tank initially contains \(20 \mathrm{L}\) of water. A solution containing \(1 \mathrm{g} / \mathrm{L}\) of chemical flows into the tank at a rate of \(3 \mathrm{L} / \mathrm{min},\) and the mixture flows out at a rate of 2 L/min. (a) Set up and solve the initial-value problem for \(A(t),\) the amount of chemical in the tank at time \(t\) (b) When does the concentration of chemical in the tank reach 0.5 g/L?