4.1E: Exercises

Exercise $\PageIndex{1}$

Tanks $T_1$ and $T_2$ contain $50$ gallons and $100$ gallons of salt solutions, respectively. A solution with $2$ pounds of salt per gallon is pumped into $T_1$ from an external source at $1$ gal/min, and a solution with $3$ pounds of salt per gallon is pumped into $T_2$ from an external source at $2$ gal/min. The solution from $T_1$ is pumped into $T_2$ at $3$ gal/min, and the solution from $T_2$ is pumped into $T_1$ at $4$ gal/min. $T_1$ is drained at $2$ gal/min and $T_2$ is drained at $1$ gal/min. Let $Q_1(t)$ and $Q_2(t)$ be the number of pounds of salt in $T_1$ and $T_2$, respectively, at time $t>0$. Derive a system of differential equations for $Q_1$ and $Q_2$. Assume that both mixtures are well stirred.

Exercise $\PageIndex{2}$

Two $500$ gallon tanks $T_1$ and $T_2$ initially contain $100$ gallons each of salt solution. A solution with $2$ pounds of salt per gallon is pumped into $T_1$ from an external source at $6$ gal/min, and a solution with $1$ pound of salt per gallon is pumped into $T_2$ from an external source at $5$ gal/min. The solution from $T_1$ is pumped into $T_2$ at $2$ gal/min, and the solution from $T_2$ is pumped into $T_1$ at $1$ gal/min. Both tanks are drained at $3$ gal/min. Let $Q_1(t)$ and $Q_2(t)$ be the number of pounds of salt in $T_1$ and $T_2$, respectively, at time $t>0$. Derive a system of differential equations for $Q_1$ and $Q_2$ that's valid until a tank is about to overflow. Assume that both mixtures are well stirred.

Answer

$Q_1'=(\mbox{rate in})_1-(\mbox{rate out})_1$ and $Q_2'=(\mbox{rate in})_2-(\mbox{rate out})_2$ .

The volumes of the solutions in $T_1$ and $T_2$ are $V_1(t)=100+2t$
and $V_2(t)=100+3t$ , respectively. $T_1$ receives salt from the
external source at the rate of $\mbox{(2 lb/gal) }\times\mbox{ (6~gal/min)}=\mbox{12 lb/min}$ , and from $T_2$ at the rate of
$\mbox{(lb/gal in }T_2)\times\mbox{ (1~gal/min) }=\displaystyle{1\over100+3t}Q_2 \mbox{ lb/min}$ . Therefore, (A) $\mbox{(rate in)}_1= 12+\displaystyle{1\over100+3t}Q_2$ . Solution leaves $T_1$ at 5~gal/min,
since 3~gal/min are drained and 2~gal/min are pumped to $T_2$ ; hence
(B) $(\mbox{rate out})_1=(\mbox{ lb/gal in T}_1)\times \mbox{(5~gal/min) } =\displaystyle{1\over100+2t}Q_1\times5=\displaystyle{5\over100+2t}Q_1$ . Now (A) and (B)
imply that (C) $Q_1'=\displaystyle{12-{5\over100+2t}Q_1+{1\over100+3t}Q_2}$ .

$T_2$ receives salt from the external source at the rate of $\mbox{(1 lb/gal) }\times\mbox{ (5~gal/min)}=\mbox{ 5 lb/min}$ , and from $T_1$
at the rate of \) \mbox{(lb/gal in }T_1)\times\mbox{ (2~gal/min)
}=\displaystyle{1\over100+2t}Q_1\times2=\displaystyle{1\over50+t}Q_1 \mbox{ lb/min}\) .
Therefore, (D) $\mbox{(rate in)}_2= 5+\displaystyle{1\over50+t}Q_1$ . Solution
leaves $T_2$ at 4~gal/min, since 3~gal/min are drained and 1~gal/min
is pumped to $T_1$ ; hence (E) $(\mbox{rate out})_2=(\mbox{ lb/gal in T}_2)\times \mbox{(4~gal/min) } =\displaystyle{1\over100+3t}Q_2\times4=\displaystyle{4\over100+3t}Q_2$ . Now (D) and (E)
imply that (F) $Q_2'=\displaystyle{5+{1\over50+t}Q_1-{4\over100+3t}Q_2}$ . Now
(C) and (F) form the desired system.
 

Exercise $\PageIndex{3}$

A mass $m_1$ is suspended from a rigid support on a spring $S_1$ with spring constant $k_1$ and damping constant $c_1$. A second mass $m_2$ is suspended from the first on a spring $S_2$ with spring constant $k_2$ and damping constant $c_2$, and a third mass $m_3$ is suspended from the second on a spring $S_3$ with spring constant $k_3$ and damping constant $c_3$. Let $y_1=y_1(t)$, $y_2=y_2(t)$, and $y_3=y_3(t)$ be the displacements of the three masses from their equilibrium positions at time $t$, measured positive upward. Derive a system of differential equations for $y_1$, $y_2$ and $y_3$, assuming that the masses of the springs are negligible and that vertical external forces $F_1$, $F_2$, and $F_3$ also act on the masses.

Exercise $\PageIndex{4}$

Let ${\bf X}=x\,{\bf i}+y\,{\bf j}+z\,{\bf k}$ be the position vector of an object with mass $m$, expressed in terms of a rectangular coordinate system with origin at Earth's center (Figure $(4.1.3)$). Derive a system of differential equations for $x$, $y$, and $z$, assuming that the object moves under Earth's gravitational force (given by Newton's law of gravitation, as in Example $(4.1.3)$ ) and a resistive force proportional to the speed of the object. Let $\alpha$ be the constant of proportionality.

Answer

$m{\bf X}''=-\alpha{\bf X'}-mgR^2\displaystyle{{\bf X}\over\|{\bf X}\|^3}$ ;
see Example~4.1.3.

Exercise $\PageIndex{5}$

Rewrite the given system as a first order system.

(a) $$\begin{array}{lcc} x''' = f(t,x,y,y')\\ y'' = g(t,y,y') \end{array}$$

(b) $$\begin{array}{lcl} u' = f(t,u,v,v',w')\\ v''=g(t,u,v,v',w)\\ w''=h(t,u,v,v',w,w')\end{array}$$

(c) $y''' = f(t,y,y',y'')$

(d) $y^{(4)} = f(t,y)$

(e) $$\begin{array}{lcc} x'' = f(t,x,y)\\ y'' = g(t,x,y) \end{array}$$

Exercise $\PageIndex{6}$

Rewrite the system Equation $(4.1.14)$ of differential equations derived in Example $(4.1.3)$ as a first order system.

Exercise $\PageIndex{7}$

Formulate a version of Euler's method (Section 3.1) for the numerical solution of the initial value problem

$$\begin{array}{rcl} y_1'&=&g_1(t,y_1,y_2),\quad y_1(t_0)=y_{10},\\ y_2'&=&g_2(t,y_1,y_2),\quad y_2(t_0)=y_{20}, \end{array}$$

on an interval $[t_0,b]$.

Exercise $\PageIndex{8}$

Formulate a version of the improved Euler method (Section 3.2) for the numerical solution of the initial value problem

$$\begin{array}{rcl} y_1'&=&g_1(t,y_1,y_2),\quad y_1(t_0)=y_{10},\\ y_2'&=&g_2(t,y_1,y_2),\quad y_2(t_0)=y_{20}, \end{array}$$

on an interval $[t_0,b]$.

Answer

$$\begin{eqnarray*} I_{1i}&=&g_1(t_i,y_{1i},y_{2i}),\\ J_{1i}&=&g_2(t_i,y_{1i},y_{2i}),\\ I_{2i}&=&g_1\left(t_i+h,y_{1i}+hI_{1i},y_{2i}+hJ_{1i}\right),\\ J_{2i}&=&g_2\left(t_i+h,y_{1i}+hI_{1i},y_{2i}+hJ_{1i}\right),\\ y_{1,i+1}&=&y_{1i}+{h\over2}(I_{1i}+I_{2i}),\\ y_{2,i+1}&=&y_{2i}+{h\over2}(J_{1i}+J_{2i}). \end{eqnarray*}$$