The Michaelis-Menten equations describe a biochemical reaction in which an enzyme E and substrate S bind to form a complex C. This complex can then either dissociate back into its original components or undergo a reaction in which a product $P$ is produced along with the free enzyme: $\mathrm{E}+\mathrm{S} \leftrightarrow \mathrm{C} \rightarrow \mathrm{E}+\mathrm{P}$This can be expressed by the differential equations

$$\begin{aligned} &\frac{d x}{d t}=-k_{f} x y M+k_{r}(1-y) M\\ &\frac{d y}{d t}=-k_{f} x y M+k_{r}(1-y) M+k_{\mathrm{cat}}(1-y) M\\ &\frac{d z}{d t}=k_{\mathrm{cat}}(1-y) M \end{aligned}$$

Where $M$ is the total number of enzymes (both free and bound), $x$ and $z$ are the numbers of the substrate and product molecules, $y$ is the fraction of the enzyme pool that is free, and the $k_i's$ are positive constants. Calculate the Jacobian matrix.

Superposition First order linear differential equations possess important superposition properties. Show the following: (a) If $y_1(t)$ and $y_2(t)$ are any two solutions of the homogeneous equation $y^{\prime}+p(t) y=0$ and if $c_1$ and $c_2$ are any two constants, then the sum $c_1 y_1(t)+c_2 y_2(t)$ is also a solution of the homogeneous equation. (b) If $y_1(t)$ is a solution of the homogeneous equation $y^{\prime}+p(t) y=0$ and $y_2(t)$ is a solution of the nonhomogeneous equation $y^{\prime}+p(t) y=g(t)$ and $c$ is any constant, then the sum $c y_1(t)+y_2(t)$ is also a solution of the nonhomogeneous equation. (c) If $y_1(t)$ and $y_2(t)$ are any two solutions of the nonhomogeneous equation $y^{\prime}+p(t) y=g(t)$, then the sum $y_1(t)+y_2(t)$ is not a solution of the nonhomogeneous equation.

Solve the following differential equations subject to the specified initial conditions.

$$\begin{array}{l}{\text { (a) } d^{2} v / d t^{2}+4 v=12, v(0)=0, d v(0) / d t=2} \\ {\text { (b) } d^{2} i / d t^{2}+5 d i / d t+4 i=8, i(0)=-1, \\ \quad d i(0) / d t=0} \\ {\text { (c) } d^{2} v / d t^{2}+2 d v / d t+v=3, v(0)=5, d v(0) / d t=1} \\ {\text { (d) } d^{2} i / d t^{2}+2 d i / d t+5 i=10, i(0)=4, \\ \quad d i(0) / d t=-2}\end{array}$$

Hemodialysis is a process by which a machine is used to filter urea and other waste products from a patient’s blood if the kidneys fail. The amount of urea within a patient during dialysis is sometimes modeled by supposing there are two compartments within the patient: the blood, which is directly filtered by the dialysis machine, and another compartment that cannot be directly filtered but that is connected to the blood. A system of two differential equations describing this is $\frac{d c}{d t}=-\frac{K}{V} c+a p-b c \quad \frac{d p}{d t}=-a p+b c$ where c and p are the urea concentrations in the blood and the inaccessible pool (in mg/mL) and all constants are positive. Suppose that $K / V=1, a=b=\frac{1}{2},$ and the initial urea concentration is $c(0)=c_{0} \text { and } p(0)=c_{0} \mathrm{mg} / \mathrm{mL}.$ Solve this initial-value problem.

A certain ecological territory contains S thousand squirrels and R thousand rabbits. Currently, there are 4,000 of each species and the growth rates of the populations with respect to time t are given by this pair of differential equations:

$$\begin{array}{l}{\frac{d R}{d t}=63 R-3 R S} \\ {\frac{d S}{d t}=26 S-R S}\end{array}$$

a. Find equilibrium populations for this model, that is, populations $R_e$ and $S_e$ that satisfy $\frac{d R}{d t}=\frac{d S}{d t}=0$ b. Note that $\frac{d R}{d S}=\frac{63 R-3 R S}{26 S-R S}.$ Separate the variables in this equation, and solve to obtain an implicit solution for the system.

Hemodialysis is a process by which a machine is used to filter urea and other waste products from a patient’s blood if the kidneys fail. The amount of urea within a patient during dialysis is sometimes modeled by supposing there are two compartments within the patient: the blood, which is directly filtered by the dialysis machine, and another compartment that cannot be directly filtered but that is connected to the blood. A system of two differential equations describing this is $\frac{d c}{d t}=-\frac{K}{V} c+a p-b c \quad \frac{d p}{d t}=-a p+b c$ where c and p are the urea concentrations in the blood and the inaccessible pool (in mg/mL) and all constants are positive. Suppose that $K / V=1, a=b=\frac{1}{2},$ and the initial urea concentration is $c(0)=c_{0} \text { and } p(0)=c_{0} \mathrm{mg} / \mathrm{mL}.$ Classify the equilibrium of this system.

Making Equations Separable Many differential equations that are not separable can be made separable by making a proper substitution. One example is the class of first-order equations with right-hand sides that are functions of the combination $y / t ( or t / y )$. Given such a DE $\frac{d y}{d t}=f\left(\frac{y}{t}\right)$ called Euler-homogeneous, let $v = y / t$. By the product rule, we deduce from $y = vt$ that $\frac{d y}{d t}=v+t \frac{d v}{d t}$ so the equation becomes $v+t \frac{d v}{d t}=f(v)$ which separates into $\frac{d t}{t}=\frac{d v}{f(v)-v}$ Apply this method to sol ve the Euler-homogeneous DE and IVP. Plot sample solutions on a direction field and discuss. $\frac{d y}{d t}=\frac{2 y^{4}+t^{4}}{t y^{3}}$

In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations. The delivery of a drug (such as an antibiotic) through an intravenous line may be modeled by the differential equation $m^{\prime}(t)+k m(t)=I$. where m(t) is the mass of the drug in the blood at time $t \geq 0$, k is a constant that describes the rate at which the drug is absorbed, and l is the infusion rate. a. Show by substitution that if the initial mass of drug in the blood is zero (m(0)=0), then the solution of the initial value problem is $m(t)=\frac{I}{k}\left(I-e^{-k t}\right)$ b. Graph the solution for l=10 m/hr and $k=0.05 hr^{-1}$. c. Evaluate $\lim _{t \rightarrow \infty} m(t)$, the steady-state drug level, and verify the result using the graph in part (b).

When two countries are in an arms race, the rate at which each one spends money on arms is determined by its own current level of spending and by its opponent's level of spending. We expect that the more a country is already spending on armaments, the less willing it will be to increase its military expenditures. On the other hand, the more a country's opponent spends on armaments, the more rapidly the country will arm. If $\$ x$ billion is the country's yearly expenditure on arms, and $\$ y$ billion is its opponent's, then the Richardson arms race model proposes that $x$ and $y$ are determined by differential equations. Suppose that for some particular arms race the equations are

$$\begin{aligned} &\frac{d x}{d t}=-0.2 x+0.15 y+20 \\ &\frac{d y}{d t}=0.1 x-0.2 y+40 . \end{aligned}$$

(f) What does this model predict will happen in the case of unilateral disarmament (one country disarms, and the other country does not)?

One morning snow began to fall at a heavy and constant rate. A snowplow started out at 8:00 A.M. At 9:00 A.M. it had traveled 2 miles. By 10:00 A.M. it had traveled 3 miles. Assuming that the snowplow removes a constant volume of snow per hour, determine the time at which it started snowing. (Hint: Let t denote the time since the snow started to fall, and let T be the time when the snowplow started out. Let x, the distance the snowplow has traveled, and h, the height of the snow, be functions of t. The assumption that a constant volume of snow per hour is removed implies that the speed of the snowplow times the height of the snow is a constant. Set up and solve differential equations involving dx/dt and dh/dt.)

In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations. The growth of cancer tumors may be modeled by the Gompertz growth equation. Let M(t) be the mass of a tumor for $t \geq 0$. The relevant initial value problem is $\frac{d M}{d t}=-r M(t) \ln \left(\frac{M(t)}{K}\right), M(0)=M_{0}$ where r and K are positive constants and $0<M_{0}<K$. a. Show by substitution that the solution of the initial value problem is $M(t)=K\left(\frac{M_{0}}{K}\right)^{\exp (-rt)}$. b. Graph the solution for $M_{0}=100$ and r=0.05. c. Using the graph in part (b), estimate $\lim _{t \rightarrow \infty} M(t)$, the limiting size of the tumor.

Let $D$ and $I$ denote the national debt and national income, and assume that both are functions of time $t$. One of several Domar debt models assumes that the time rates of change of $D$ and $I$ are both proportional to $I$, so that

$$\frac{d D}{d t}=a I \text { and } \frac{d I}{d t}=b I$$

Suppose $I(0)=I_0$ and $D(0)=D_0$.

a. Solve both of these differential equations and express $D(t)$ and $I(t)$ in terms of $a, b, I_0$, and $D_0$.

b. The economist, Evsey Domar, who first studied this model, was interested in the ratio of national debt to national income. Determine what happens to this ratio in the long run by computing the limit

$$\lim _{t \rightarrow \infty} \frac{D(t)}{I(t)}$$