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Section Chapter 12 Exercises
Reading Questions α―β
β Quick-Answer Questions
1. True-False.
(a) πβ Which Variable Drives Which?
\begin{align*}
\frac{dx}{dt} \amp = x - y\\
\frac{dy}{dt} \amp = -2y
\end{align*}
Select all true statements:
\(x\) evolves independently of
\(y\text{.}\)
\(y\) depends on
\(x\text{.}\)
This is an uncoupled system.
Not quite β notice that \(x\) shows up in the \(dy/dt\) equation.
You must solve for
\(y\) before you can solve for
\(x\text{.}\)
Actually, you can solve \(x\) first since itβs independent.
(b) Interdependence.
What makes a system fully coupled? How does that affect how we approach the problem?
(c) πβ Classify the System.
Match each system to the correct classification.
Assume both \(x\) and \(y\) are functions of \(t\text{.}\)
Systems are classified by whether variables influence each other and how. Look for dependencies in the right-hand sides.
\begin{align*}
x' \amp = x,\\
y' \amp = y
\end{align*}
\begin{align*}
x' \amp = \cos(t),\\
y' \amp = y
\end{align*}
Uncoupled
\begin{align*}
x' \amp = x,\\
y' \amp = x + y
\end{align*}
\begin{align*}
x' \amp = x,\\
y' \amp = t + x
\end{align*}
Partially Coupled
\begin{align*}
x' \amp = x + y,\\
y' \amp = x - y
\end{align*}
Fully Coupled
(d) πβ Identifying Features.
Select all statements that are true about partially coupled systems.
One equation is independent and can be solved first.
Yes, this is a defining feature of partial coupling.
Both equations must be solved simultaneously.
No, only fully coupled systems require that.
The dependent equation uses the solution of the independent one.
Exactly. You substitute \(x(t)\) into the second equation to solve for \(y(t)\text{.}\)
Partial coupling means both variables evolve independently.
That describes an uncoupled system, not a partially coupled one.
Only one of the equations involves both variables.
Correct. The coupling only appears in one direction.
The second variable can be solved without knowing the first.
Nope, you need \(x(t)\) to solve for \(y(t)\text{.}\)
(e) πβ Select the True Statements.
Which of the following statements are true for the system:
\begin{align*}
\frac{dx}{dt} \amp = -x + 1 \\
\frac{dy}{dt} \amp = -2y
\end{align*}
The rate of change of \(x\) depends on \(y\text{.}\)
The variable \(x\) has no effect on how \(y\) changes.
The rate of change of \(y\) depends only on \(y\text{.}\)
This is an example of a coupled system.
This system has two independent variables.
(f) πβ Independent Solutions.
In an uncoupled system, how do we find the solution for the whole system?
Solve each equation separately, then combine the answers into a pair
\((x(t), y(t))\text{.}\)
Rewrite the system as a second-order equation for just one variable.
You could do this in some cases, but for uncoupled systems itβs simpler to solve each one directly.
Use Eulerβs Method only β they cannot be solved exactly.
Uncoupled systems are straightforward to solve exactly.
Exercises Exercises
Exercise Group. Solve the following systems of differential equations.
1. \(\ds \frac{dx}{dt} = -x+y, \ds \frac{dy}{dt} = 2x,\quad x(0) = 0,\quad y(0) = 1 \) 2. \(\ds \frac{dx}{dt} = x - 2y, \ds \frac{dy}{dt} = 5x - y,\quad x(0) = -1,\quad y(0) = 2 \) 3. \(y' - 2x = 1, \ds x' + y' - 3x - 3y = 2,\quad x(0) = 0,\quad y(0) = 0 \)
Exercise Group. Suppose a mixture containing 0.3 kg of sugar per liter runs into a tank initially filled with 400 L of water containing 2 kg of sugar. The liquid enters at 10 L/min,the mixture is kept uniform by stirring, and the mixture flows out at the same rate.
4. \(\ds t \) minutes.5. 6. concentration of sugar in the tank after a long time. Does your answer make sensein terms of the physical scenario? Explain.
Exercise Group. 7. \(\ds t \) minutes.
8. From a Second-Order Equation to a System.
Rewrite the second-order equation
\(y'' - y = 0\) as a system of first-order equations.
You have attempted
of
activities on this page.
