Problem 34
An endowment is an investment account in which the balance ideally remains constant and withdrawals are made on the interest earned by the account. Such an account may be modeled by the initial value problem \(B^{\prime}(t)=r B-m,\) for \(t \geq 0,\) with \(B(0)=B_{0} .\) The constant \(r>0\) reflects the annual interest rate, \(m>0\) is the annual rate of withdrawal, \(B_{0}\) is the initial balance in the account, and \(t\) is measured in years. a. Solve the initial value problem with \(r=0.05, m=\$ 1000 /\) year, and \(B_{0}=\$ 15,000 .\) Does the balance in the account increase or decrease? b. If \(r=0.05\) and \(B_{0}=\$ 50,000,\) what is the annual withdrawal rate \(m\) that ensures a constant balance in the account? What is the constant balance?
Problem 35
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 function, be sure to indicate which function corresponds to the solution of the initial value problem. $$u^{\prime}(x)=\csc u \cos \frac{x}{2}, u(\pi)=\frac{\pi}{2}$$
Problem 35
Solving initial value problems Solve the following initial value problems. $$y^{\prime}(x)=3 x^{2}-3 x^{-4}, y(1)=0$$
Problem 35
Determine whether the following statements are true and give an explanation or counterexample. a. The general solution of \(y^{\prime}(t)=2 y-18\) is \(y(t)=2 e^{2 t}+9\) b. If \(k>0\) and \(b>0,\) then \(y(t)=0\) is never a solution of \(y^{\prime}(t)=k y-b\) c. The equation \(y^{\prime}(t)=t y(t)+3\) is separable and can be solved using the methods of this section. d. According to Newton's Law of Cooling, the temperature of a hot object will reach the ambient temperature after a finite amount of time.
Problem 35
Properties of stirred tank solutions a. Show that for general positive values of \(R, V, C_{i},\) and \(m_{0},\) the solution of the initial value problem $$m^{\prime}(t)=-\frac{R}{V} m(t)+C_{i} R, \quad m(0)=m_{0}$$is \(m(t)=\left(m_{0}-C_{i} V\right) e^{-R t / V}+C_{i} V\) b. Verify that \(m(0)=m_{0}\) c. Evaluate \(\lim _{t \rightarrow \infty} m(t)\) and give a physical interpretation of the result. d. Suppose \(m_{0}\) and \(V\) are fixed. Describe the effect of increasing \(R\) on the graph of the solution.
Problem 36
Solve the equation \(y^{\prime}(t)=k y+b\) in the case that \(k y+b<0\) and verify that the general solution is \(y(t)=C e^{k t}-\frac{b}{k}\)
Problem 36
A physiological model A common assumption in modeling drug assimilation is that the blood volume in a person is a single compartment that behaves like a stirred tank. Suppose the blood volume is a four-liter tank that initially has a zero concentration of a particular drug. At time \(t=0,\) an intravenous line is inserted into a vein (into the tank) that carries a drug solution with a concentration of \(500 \mathrm{mg} / \mathrm{L} .\) The inflow rate is \(0.06 \mathrm{L} / \mathrm{min} .\) Assume the drug is quickly mixed thoroughly in the blood and that the volume of blood remains constant. a. Write an initial value problem that models the mass of the drug in the blood, for \(t \geq 0\) b. Solve the initial value problem, and graph both the mass of the drug and the concentration of the drug. c. What is the steady-state mass of the drug in the blood? d. After how many minutes does the drug mass reach \(90 \%\) of its steady-state level?
Problem 36
Solving initial value problems Solve the following initial value problems. $$y^{\prime}(x)=4 \sec ^{2} 2 x, y(0)=8$$
Problem 36
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 function, be sure to indicate which function corresponds to the solution of the initial value problem. $$y y^{\prime}(x)=\frac{2 x}{\left(2+y^{2}\right)^{2}}, y(1)=-1$$
Problem 37
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 function, be sure to indicate which function corresponds to the solution of the initial value problem. $$y^{\prime}(x) \sqrt{y+4}=\sqrt{x+1}, y(3)=5$$
