Chapter 2: Problem 7
Draw a direction field for the given differential equation and state whether you think that the solutions are converging or diverging. $$ y^{\prime}=(4-t y) /\left(1+y^{2}\right) $$
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A body of constant mass \(m\) is projected vertically upward with an initial velocity \(v_{0}\) in a medium offering a resistance \(k|v|,\) where \(k\) is a constant. Neglect changes in the gravitational force. $$ \begin{array}{l}{\text { (a) Find the maximum height } x_{m} \text { attained by the body and the time } t_{m} \text { at which this }} \\ {\text { maximum height is reached. }} \\ {\text { (b) Show that if } k v_{0} / m g<1, \text { then } t_{m} \text { and } x_{m} \text { can be expressed as }}\end{array} $$ $$ \begin{array}{l}{t_{m}=\frac{v_{0}}{g}\left[1-\frac{1}{2} \frac{k v_{0}}{m g}+\frac{1}{3}\left(\frac{k v_{0}}{m g}\right)^{2}-\cdots\right]} \\\ {x_{m}=\frac{v_{0}^{2}}{2 g}\left[1-\frac{2}{3} \frac{k r_{0}}{m g}+\frac{1}{2}\left(\frac{k v_{0}}{m g}\right)^{2}-\cdots\right]}\end{array} $$ $$ \text { (c) Show that the quantity } k v_{0} / m g \text { is dimensionless. } $$
solve the given initial value problem and determine how the interval in which the solution exists depends on the initial value \(y_{0}\). $$ y^{\prime}=2 t y^{2}, \quad y(0)=y_{0} $$
Involve equations of the form \(d y / d t=f(y) .\) In each problem sketch the
graph of \(f(y)\) versus \(y\), determine the critical (equilibrium) points, and
classify each one as asymptotically stable, unstable, or semistable (see
Problem 7 ).
$$
d y / d t=y^{2}\left(4-y^{2}\right), \quad-\infty
Use the technique discussed in Problem 20 to show that the approximation obtained by the Euler method converges to the exact solution at any fixed point as \(h \rightarrow 0 .\) $$ y^{\prime}=2 y-1, \quad y(0)=1 \quad \text { Hint: } y_{1}=(1+2 h) / 2+1 / 2 $$
Involve equations of the form \(d y / d t=f(y)\). In each problem sketch the
graph of \(f(y)\) versus \(y,\) determine the critical (equilibrium) points, and
classify each one as asymptotically stable or unstable.
$$
d y / d t=-2(\arctan y) /\left(1+y^{2}\right), \quad-\infty
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