Chapter 2

A small metal bar is removed from an oven whose temperature is a constant \(300^{\circ} \mathrm{F}\) into a room whose temperature is a constant \(70^{\circ} \mathrm{F}\). Simultaneously, an identical metal bar is removed from the room and placed into the oven. Assume that time \(t\) is measured in minutes. Discuss: Why is there a future value of time, call it \(t^{*}>0\), at which the temperature of each bar is the same?

Without the use of technology, how would you solve $$ (\sqrt{x}+x) \frac{d y}{d x}=\sqrt{y}+y ? $$ Carry out your ideas.

Cooling and Warming A small metal bar is removed from an oven whose temperature is a constant \(300^{\circ} \mathrm{F}\) into a room whose temperature is a constant \(70^{\circ} \mathrm{F}\). Simultaneously, an identical metal bar is removed from the room and placed into the oven. Assume that time \(t\) is measured in minutes. Discuss: Why is there a future value of time, call it \(t^{*}>0\), at which the temperature of each bar is the same?

Suppose \(P(x)\) is continuous on some interval \(I\) and \(a\) is a number in \(I\). What can be said about the solution of the initial-value problem \(y^{\prime}+P(x) y=0, y(a)=0 ?\)

Find a function whose square plus the square of its derivative is \(1 .\)

Radioactive Decay Series The following system of differential equations is encountered in the study of the decay of a special type of radioactive series of elements: $$ \begin{aligned} &\frac{d x}{d t}=-\lambda_{1} x \\ &\frac{d y}{d t}=\lambda_{1} x-\lambda_{2} y \end{aligned} $$ where \(\lambda_{1}\) and \(\lambda_{2}\) are constants. Discusshow to solve this system subject to \(x(0)=x_{0}, y(0)=y_{0}\). Carry out your ideas.

Heart Pacemaker A heart pacemaker consists of a switch, a battery of constant voltage \(E_{0}\), a capacitor with constant capacitance \(C\), and the heart as a resistor with constant resistance \(R\). When the switch is closed, the capacitor charges; when the switch is open, the capacitor discharges, sending an electrical stimulus to the heart. During the time the heart is being stimulated, the voltage \(E\) across the heart satisfies the linear differential equation $$ \frac{d E}{d t}=-\frac{1}{R C} E $$ Solve the \(\mathrm{DE}\) subject to \(E(4)=E_{0}\)

(a) Express the solution of the initial-value problem \(y^{\prime}-2 x y=-1, y(0)=\sqrt{\pi / 2}\), in terms of erfc \((x)\) (b) Use tables or a CAS to find the value of \(y(2)\). Use a CAS to graph the solution curve for the IVP on the interval \((-\infty, \infty)\)

(a) Use a CAS and the concept of level curves to plot representative graphs of members of the family of solutions of the differential equation \(\frac{d y}{d x}=-\frac{8 x+5}{3 y^{2}+1} .\) Experiment with different numbers of level curves as well as various rectangular regions defined by \(a \leq x \leq b, c \leq y \leq d\). (b) On separate coordinate axes plot the graphs of the particular solutions corresponding to the initial conditions: \(y(0)=-1 ; y(0)=2 ; y(-1)=4 ; y(-1)=-3\)

(a) The sine integral function is defined by \(\operatorname{Si}(x)=\) \(\int_{0}^{x}(\sin t / t) d t\), where the integrand is defined to be 1 at \(t=0\). Express the solution \(y(x)\) of the initial-value problem \(x^{3} y^{\prime}+2 x^{2} y=10 \sin x, y(1)=0\), in terms of \(\operatorname{Si}(x)\). (b) Use a CAS to graph the solution curve for the IVP for \(x>0 .\) (c) Use a CAS to find the value of the absolute maximum of the solution \(y(x)\) for \(x>0\).

(a) Find an implicit solution of the IVP $$ (2 y+2) d y-\left(4 x^{3}+6 x\right) d x=0, \quad y(0)=-3 $$ (b) Use part (a) to find an explicit solution \(y=\phi(x)\) of the IVP. (c) Consider your answer to part (b) as a function only. Use a graphing utility or a CAS to graph this function, and then use the graph to estimate its domain. (d) With the aid of a root-finding application of a CAS, determine the approximate largest interval \(I\) of definition of the solution \(y=\phi(x)\) in part (b). Use a graphing utility or a CAS to graph the solution curve for the IVP on this interval.

(a) The Fresnel sine integral is defined by \(S(x)=\) \(\int_{0}^{x} \sin \left(\pi t^{2} / 2\right) d t\). Express the solution \(y(x)\) of the initialvalue problem \(y^{\prime}-\left(\sin x^{2}\right) y=0, y(0)=5\), in terms of \(S(x)\) (b) Use a CAS to graph the solution curve for the IVP on \((-\infty, \infty)\) (c) It is known that \(S(x) \rightarrow \frac{1}{2}\) as \(x \rightarrow \infty\) and \(S(x) \rightarrow-\frac{1}{2}\) as \(x \rightarrow-\infty\). What does the solution \(y(x)\) approach as \(x \rightarrow \infty ?\) As \(x \rightarrow-\infty ?\) (d) Use a CAS to find the values of the absolute maximum and the absolute minimum of the solution \(y(x)\).