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*}