91Ó°ÊÓ

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the Runge-Kutta method to approximate this solution on the interval \([0,0.5]\) with step size \(h=0.25 .\) Construct a table showing five-decimal-place values of the approximate solution and actual solution at the points \(x=0.25\) and \(0.5\). $$ y^{\prime}=y+1, y(0)=1 ; y(x)=2 e^{x}-1 $$

An initial value problem and its exact solution \(y(x)\) are given. Apply Euler's method twice to approximate to this solution on the interval \(\left[0, \frac{1}{2}\right]\), first with step size \(h=0.25\), then with step size \(h=0.1 .\) Compare the three-decimal-place values of the two approximations at \(x=\frac{1}{2}\) with the value \(y\left(\frac{1}{2}\right)\) of the actual solution. \(y^{\prime}=2 x y^{2}, y(0)=1 ; y(x)=\frac{1}{1-x^{2}}\)

The time rate of change of an alligator population \(P\) in a swamp is proportional to the square of \(P .\) The swamp contained a dozen alligators in 1988 , two dozen in 1998 . When will there be four dozen alligators in the swamp? What happens thereafter?

Use a computer system or graphing calculator to plot a slope field and/or enough solution curves to indicate the stability or instability of each critical point of the given differential equation. $$ \frac{d x}{d t}=x^{3}\left(x^{2}-4\right) $$

Suppose that a projectile is fired straight upward from the surface of the earth with initial velocity \(v_{0}<\sqrt{2 G M / R}\) Then its height \(y(t)\) above the surface satisfies the initial value problem $$ \frac{d^{2} y}{d t^{2}}=-\frac{G M}{(y+R)^{2}} ; \quad y(0)=0, \quad y^{\prime}(0)=v_{0} $$ Substitute \(d v / d t=v(d v / d y)\) and then integrate to obtain $$ v^{2}=v_{0}^{2}-\frac{2 G M y}{R(R+y)} $$ for the velocity \(v\) of the projectile at height \(y\). What maximum altitude does it reach if its initial velocity is \(1 \mathrm{~km} / \mathrm{s}\) ?