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# 10.1E: Introduction to Systems of Differential Equations (Exercises)

• • Contributed by William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University
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## Q10.1.1

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.

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.

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.

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 10.1.3). 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 10.1.3) and a resistive force proportional to the speed of the object. Let $$\alpha$$ be the constant of proportionality.

5. Rewrite the given system as a first order system.

1. $$\begin{array}{lcc} x''' = f(t,x,y,y')\$4pt] y'' = g(t,y,y') \end{array}$$ 2. $$\begin{array}{lcl} u' = f(t,u,v,v',w')\\[4pt] v''=g(t,u,v,v',w) \\[4pt] w''=h(t,u,v,v',w,w')\end{array}$$ 3. $$y''' = f(t,y,y',y'')$$ 4. $$y^{(4)} = f(t,y)$$ 5. $$\begin{array}{lcc} x'' = f(t,x,y)\\[4pt] y'' = g(t,x,y) \end{array}$$ 6. Rewrite the system Equation 10.1.14 of differential equations derived in Example 10.1.3 as a first order system. 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}\nonumber$ on an interval $$[t_0,b]$$.

8. Formulate a version of the improved Euler method (Section 3.2) for the numerical solution of the initial value problem $\begin{array}{rcl} y_1'&=&g_1(t,y_1,y_2),\quad y_1(t_0)=y_{10},\\ y_2'&=&g_2(t,y_1,y_2),\quad y_2(t_0)=y_{20}, \end{array}\nonumber$ on an interval $$[t_0,b]$$.