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Chapter 7: Problem 14
Evaluate the following integrals or state that they diverge. $$\int_{2}^{\infty} \frac{d y}{y \ln y}$$
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An object in free fall may be modeled by assuming that the only forces at work
are the gravitational force and resistance (friction due to the medium in
which the object falls). By Newton's second law (mass \(\times\) acceleration
\(=\) the sum of the external forces), the velocity of the object satisfies the
differential equation $$m \quad \cdot \quad v^{\prime}(t)=m g+f(v)$$, where
\(f\) is a function that models the resistance and the positive direction is
downward. One common assumption (often used for motion in air) is that
\(f(v)=-k v^{2},\) where \(k>0\) is a drag coefficient.
a. Show that the equation can be written in the form \(v^{\prime}(t)=\) \(g-a
v^{2},\) where \(a=k / m\).
b. For what (positive) value of \(v\) is \(v^{\prime}(t)=0 ?\) (This equilibrium
solution is called the terminal velocity.)
c. Find the solution of this separable equation assuming \(v(0)=0\) and
\(0
Evaluate the following integrals. Assume a and b are real numbers and \(n\) is an integer. $$\int \frac{x}{a x+b} d x \text { (Use } u=a x+b$$
For what values of \(p\) does the integral \(\int_{2}^{\infty} \frac{d x}{x \ln ^{p} x}\) exist and what is its value (in terms of \(p\) )?
A differential equation of the form \(y^{\prime}(t)=F(y)\) is said to be autonomous (the function \(F\) depends only on \(y\) ). The constant function \(y=y_{0}\) is an equilibrium solution of the equation provided \(F\left(y_{0}\right)=0\) (because then \(y^{\prime}(t)=0,\) and the solution remains constant for all \(t\) ). Note that equilibrium solutions correspond to horizontal line segments in the direction field. Note also that for autonomous equations, the direction field is independent of \(t\). Consider the following equations. a. Find all equilibrium solutions. b. Sketch the direction field on either side of the equilibrium solutions for \(t \geq 0\). c. Sketch the solution curve that corresponds to the initial condition \(y(0)=1\). $$y^{\prime}(t)=y(y-3)$$
The growth of cancer tumors may be modeled by the Gompertz growth equation.
Let \(M(t)\) be the mass of the tumor for \(t \geq 0 .\) The relevant initial
value problem is $$\frac{d M}{d t}=-a M \ln \frac{M}{K}, \quad M(0)=M_{0}$$,
where \(a\) and \(K\) are positive constants and \(0
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