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.

 

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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.

 

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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.

 

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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.

 

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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}

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Exercise \(\PageIndex{6}\)

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

 

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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 \([t_0,b]\).

 

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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 \([t_0,b]\).

 

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