4.1E: Exercises

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Page ID: 17859

$trench-1995.jpg$
Contributed by William F. Trench
Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University

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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.

Answer: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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.

Answer: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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}

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

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

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{6}$

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

Answer: Add texts here. Do not delete this text first.

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 $[t_0,b]$.

Answer: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{1}\)

Exercise \(\PageIndex{2}\)

Exercise \(\PageIndex{3}\)

Exercise \(\PageIndex{4}\)

Exercise \(\PageIndex{5}\)

Exercise \(\PageIndex{6}\)

Exercise \(\PageIndex{7}\)

Exercise \(\PageIndex{8}\)