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Chapter 8: Problem 2
Is the equation \(t^{2} y^{\prime}(t)=\frac{t+4}{y^{2}}\) separable?
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Explain how a stirred tank reaction works.
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 lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations. a. Find the equilibrium solutions. b. Sketch the direction field, 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 Gompertz growth equation is often used to model the growth of tumors. Let \(M(t)\) be the mass of a tumor at time \(t \geq 0 .\) The relevant initial value problem is $$ \frac{d M}{d t}=-r M \ln \left(\frac{M}{K}\right), M(0)=M_{0} $$ a. Graph the growth rate function \(R(M)=-r M \ln \left(\frac{M}{K}\right)\) (which equals \(M^{\prime}(t)\) ) assuming \(r=1\) and \(K=4 .\) For what values of \(M\) is the growth rate positive? For what value of \(M\) is the growth rate a maximum? b. Solve the initial value problem and graph the solution for \(r=1, K=4,\) and \(M_{0}=1 .\) Describe the growth pattern of the tumor. Is the growth unbounded? If not, what is the limiting size of the tumor? c. In the general solution, what is the meaning of \(K ?\) where \(r\) and \(K\) are positive constants and \(0 < M_{0} < K\)
Make a sketch of the population function (as a function of time) that results from the following growth rate functions. Assume the population at time \(t=0\) begins at some positive value.
Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to indicate which curve corresponds to the solution of the initial value problem. $$y^{\prime}(x)=\sqrt{\frac{x+1}{y+4}}, y(3)=5$$
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