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Involve exponential growth. Suppose a bacterial culture initially has 100 cells. After 2 hours, the population has increased to \(400 .\) Find an equation for the population at any time. What will the population be after 8 hours?

Involve exponential growth. A bacterial culture grows exponentially with growth constant 0.12 hour \(^{-1} .\) Find its doubling time.

Use Euler's method with \(h=0.1\) and \(h=0.05\) to approximate \(y(1)\) and \(y(2) .\) Show the first two steps by hand. $$y^{\prime}=2 x y, y(0)=1$$

The differential equation is separable. Find the general solution, in an explicit form if possible. Sketch several members of the family of solutions. $$y^{\prime}=\frac{6 x^{2}}{y\left(1+x^{3}\right)}$$

The differential equation is separable. Find the general solution, in an explicit form if possible. Sketch several members of the family of solutions. $$y^{\prime}=\frac{3 x}{y+1}$$

Involve exponential growth. A bacterial culture grows exponentially with growth constant 0.12 hour \(^{-1} .\) Find its doubling time.

Use Euler's method with \(h=0.1\) and \(h=0.05\) to approximate \(y(1)\) and \(y(2) .\) Show the first two steps by hand. $$y^{\prime}=x / y, y(0)=2$$

The differential equation is separable. Find the general solution, in an explicit form if possible. Sketch several members of the family of solutions. $$y^{\prime}=\frac{2 x e^{y}}{y e^{x}}$$

Use Euler's method with \(h=0.1\) and \(h=0.05\) to approximate \(y(1)\) and \(y(2) .\) Show the first two steps by hand. $$y^{\prime}=4 y-y^{2}, y(0)=1$$

Involve exponential growth. Suppose that a population of \(E .\) coli doubles every 20 minutes. A treatment of the infection removes \(90 \%\) of the \(E .\) coli present and is timed to accomplish the following. The population starts at size \(10^{8},\) grows for \(T\) minutes, the treatment is applied and the population returns to size \(10^{8} .\) Find the time \(T\)