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Chapter 8: Problem 21

Find the equilibrium solution of the following equations, make a sketch of the direction field, for \(t \geq 0,\) and determine whether the equilibrium solution is stable. The direction field needs to indicate only whether solutions are increasing or decreasing on either side of the equilibrium solution. $$u^{\prime}(t)+7 u+21=0$$

Short Answer

Expert verified
Answer: The equilibrium solution is \(u(t) = -3\), and it is stable.

Step by step solution

01

Find the equilibrium solution

To find the equilibrium solution, we have to set the derivative equal to zero: $$u^{\prime}(t) + 7u + 21 = 0.$$ The equilibrium solution satisfies \(u^{\prime}(t) = 0,\) so: $$7u + 21 = 0.$$ Then, solving for \(u,\) we get: $$u(t) = -\frac{21}{7} = -3.$$ So, the equilibrium solution is \(u(t) = -3.\)
02

Determine the behavior of solutions

We have the differential equation: $$u^{\prime}(t) + 7u + 21 = 0.$$ If \(u(t) > -3,\) the term \(7u + 21\) becomes positive and \(u^{\prime}(t) < 0,\) meaning solutions are decreasing. If \(u(t) < -3,\) the term \(7u + 21\) becomes negative and \(u^{\prime}(t) > 0,\) meaning solutions are increasing.
03

Sketch the direction field

Based on the information from step 2, we can sketch the direction field: 1. For \(u(t) > -3,\) all arrows in the direction field should point downwards. 2. For \(u(t) = -3,\) all arrows in the direction field should be horizontal. 3. For \(u(t) < -3,\) all arrows in the direction field should point upwards. The direction field will look something like this: ``` u ^ | \ \ -------->--------- / / ---> t ```
04

Determine the stability of the equilibrium solution

Based on the direction field, we can see that when a solution starts above the equilibrium solution, it goes towards the equilibrium solution, and when it starts below the equilibrium solution, it also goes towards the equilibrium solution. Therefore, the equilibrium solution \(u(t) = -3\) is stable.

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Most popular questions from this chapter

A differential equation of the form \(y^{\prime}(t)=f(y)\) is said to be autonomous (the function \(f\) depends only on \(y\) ). The constant function \(y=y_{0}\) is an equilibrium solution of the equation provided \(f\left(y_{0}\right)=0\) (because then \(y^{\prime}(t)=0\) and the solution remains constant for all \(t\) ). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations. a. Find the equilibrium solutions. b. Sketch the direction field, for \(t \geq 0\). c. Sketch the solution curve that corresponds to the initial condition \(y(0)=1\). $$y^{\prime}(t)=y(y-3)(y+2)$$

Consider the first-order initial value problem \(y^{\prime}(t)=a y+b, y(0)=A,\) for \(t \geq 0,\) where \(a, b,\) and \(A\) are real numbers. a. Explain why \(y=-b / a\) is an equilibrium solution and corresponds to a horizontal line in the direction field. b. Draw a representative direction field in the case that \(a>0\) Show that if \(A>-b / a,\) then the solution increases for \(t \geq 0\) and if \(A<-b / a,\) then the solution decreases for \(t \geq 0\). c. Draw a representative direction field in the case that \(a<0\) Show that if \(A>-b / a,\) then the solution decreases for \(t \geq 0\) and if \(A<-b / a,\) then the solution increases for \(t \geq 0\).

Determine whether the following equations are separable. If so, solve the initial value problem. $$\sec t y^{\prime}(t)=1, y(0)=1$$

Write a logistic equation with the following parameter values. Then solve the initial value problem and graph the solution. Let \(r\) be the natural growth rate, \(K\) the carrying capacity, and \(P_{0}\) the initial population. $$r=0.2, K=300, P_{0}=50$$

a. Show that for general positive values of \(R, V, C_{i},\) and \(m_{0},\) the solution of the initial value problem $$m^{\prime}(t)=-\frac{R}{V} m(t)+C_{i} R, \quad m(0)=m_{0}$$ is \(m(t)=\left(m_{0}-C_{i} V\right) e^{-R t / V}+C_{i} V\) b. Verify that \(m(0)=m_{0}\) c. Evaluate \(\lim m(t)\) and give a physical interpretation of the result. d. Suppose \(^{t} \vec{m}_{0}^{\infty}\) and \(V\) are fixed. Describe the effect of increasing \(R\) on the graph of the solution.
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