Chapter 9

In Exercises \(1-8\) a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) c. Sketch several solution curves. $$ \frac{d y}{d x}=(y+2)(y-3) $$

Solve the differential equations. \(x \frac{d y}{d x}+y=e^{x}, \quad x>0\)

In Exercises \(1-6,\) use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places. $$ y^{\prime}=1-\frac{y}{x}, \quad y(2)=-1, \quad d x=0.5 $$

In Exercises \(1-8\) a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) c. Sketch several solution curves. $$ \frac{d y}{d x}=y^{2}-4 $$

Solve the differential equations. \(e^{x} \frac{d y}{d x}+2 e^{x} y=1\)

In Exercises \(1-6,\) use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places. $$ y^{\prime}=x(1-y), \quad y(1)=0, \quad d x=0.2 $$

In Exercises 1 and \(2,\) show that each function \(y=f(x)\) is a solution of the accompanying differential equation. $$ y^{\prime}=y^{2} $$ $$ \text { a. }y=-\frac{1}{x} \quad \text { b. } y=-\frac{1}{x+3} \quad \text { c. } y=-\frac{1}{x+C} $$

In Exercises \(1-8\) a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) c. Sketch several solution curves. $$ \frac{d y}{d x}=y^{3}-y $$

Solve the differential equations. \(x y^{\prime}+3 y=\frac{\sin x}{x^{2}}, \quad x>0\)

In Exercises \(1-6,\) use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places. $$ y^{\prime}=2 x y+2 y, \quad y(0)=3, \quad d x=0,2 $$

In Exercises 3 and \(4,\) show that the function \(y=f(x)\) is a solution of the given differential equation. $$ y=\frac{1}{x} \int_{1}^{x} \frac{e^{t}}{t} d t, \quad x^{2} y^{\prime}+x y=e^{x} $$

In Exercises \(1-8\) a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) c. Sketch several solution curves. $$ \frac{d y}{d x}=y^{2}-2 y $$

Solve the differential equations. \(y^{\prime}+(\tan x) y=\cos ^{2} x, \quad-\pi / 2< x<\pi / 2\)

In Exercises \(1-6,\) use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places. $$ y^{\prime}=y^{2}(1+2 x), \quad y(-1)=1, \quad d x=0.5 $$

In Exercises 3 and \(4,\) show that the function \(y=f(x)\) is a solution of the given differential equation. $$ y=\frac{1}{\sqrt{1+x^{4}}} \int_{1}^{x} \sqrt{1+t^{4}} d t, \quad y^{\prime}+\frac{2 x^{3}}{1+x^{4}} y=1 $$

In Exercises \(1-8\) a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) c. Sketch several solution curves. $$ y^{\prime}=\sqrt{y}, \quad y > 0 $$

Solve the differential equations. \(x \frac{d y}{d x}+2 y=1-\frac{1}{x}, \quad x>0\)

In Exercises \(1-6,\) use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places. $$ y^{\prime}=2 x e^{x^{2}}, \quad y(0)=2, \quad d x=0.1 $$

In Exercises \(5-8,\) show that each function is a solution of the given initial value problem. (TABLE NOT COPY) $$ y^{\prime}+y=\frac{2}{1+4 e^{2 x}} \quad y(-\ln 2)=\frac{\pi}{2} \quad y=e^{-x} \tan ^{-1}\left(2 e^{x}\right) $$

Gorilla population A certain wild animal preserve can support no more than 250 lowland gorillas. Twenty-eight gorillas were known to be in the preserve in \(1970 .\) Assume that the rate of growth of the population is $$ \frac{d P}{d t}=0.0004(250-P) P $$ where time \(t\) is in years. $$ \begin{array}{llllll}{t(\text { sec })} & {s(\mathbf{m})} & {t(\text { sec })} & {s(\mathbf{m})} & {t(\text { sec })} & {s(\mathbf{m})} \\ \hline 0 & {0} & {1.5} & {0.89} & {3.1} & {1.30} \\ {0.1} & {0.07} & {1.7} & {0.97} & {3.3} & {1.31} \\ {0.3} & {0.22} & {1.9} & {1.05} & {3.5} & {1.32} \\ {0.5} & {0.36} & {2.1} & {1.11} & {3.7} & {1.32} \\ {0.7} & {0.49} & {2.3} & {1.17} & {3.9} & {1.32} \\ {0.9} & {0.60} & {2.5} & {1.22} & {4.1} & {1.32} \\ {1.1} & {0.71} & {2.7} & {1.25} & {4.3} & {1.32} \\ {1.3} & {0.81} & {2.9} & {1.28} & {4.5} & {1.32}\end{array} $$ a. Find a formula for the gorilla population in terms of \(t .\) b. How long will it take for the gorilla population to reach the carrying capacity of the preserve?

In Exercises \(1-8\) a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) c. Sketch several solution curves. $$ y^{\prime}=y-\sqrt{y}, \quad y>0 $$

Solve the differential equations. \((1+x) y^{\prime}+y=\sqrt{x}\)

In Exercises \(1-6,\) use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places. $$ y^{\prime}=y+e^{x}-2, \quad y(0)=2, \quad d x=0.5 $$

In Exercises \(5-8,\) show that each function is a solution of the given initial value problem. (TABLE NOT COPY) $$ y^{\prime}=e^{-x^{2}}-2 x y \quad y(2)=0 \quad y=(x-2) e^{-x^{2}} $$

Pacific halibut fishery The Pacific halibut fishery has been modeled by the logistic equation $$ \frac{d y}{d t}=r(M-y) y $$ where \(y(t)\) is the total weight of the halibut population in kilograms at time \(t\) (measured in years), the carrying capacity is estimated to be \(M=8 \times 10^{7} \mathrm{kg},\) and \(r=0.08875 \times 10^{-7}\) per year. a. If \(y(0)=1.6 \times 10^{7} \mathrm{kg}\) , what is the total weight of the halibut population after 1 year? b. When will the total weight in the halibut fishery reach \(4 \times 10^{7} \mathrm{kg}\) ?

In Exercises \(1-8\) a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) c. Sketch several solution curves. $$ y^{\prime}=(y-1)(y-2)(y-3) $$

Solve the differential equations. \(2 y^{\prime}=e^{x / 2}+y\)

Use the Euler method with \(d x=0.2\) to estimate \(y(1)\) if \(y^{\prime}=y\) and \(y(0)=1 .\) What is the exact value of \(y(1) ?\)

In Exercises \(5-8,\) show that each function is a solution of the given initial value problem. (TABLE NOT COPY) $$ \begin{array}{l}{x y^{\prime}+y=-\sin x, \quad y\left(\frac{\pi}{2}\right)=0 \quad y=\frac{\cos x}{x}} \\ {x > 0}\end{array} $$

In Exercises \(1-8\) a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) c. Sketch several solution curves. $$ y^{\prime}=y^{3}-y^{2} $$

Solve the differential equations. \(e^{2 x} y^{\prime}+2 e^{2 x} y=2 x\)

Use the Euler method with \(d x=0.2\) to estimate \(y(2)\) if \(y^{\prime}=y / x\) and \(y(1)=2 .\) What is the exact value of \(y(2) ?\)

In Exercises \(5-8,\) show that each function is a solution of the given initial value problem. (TABLE NOT COPY) $$ \begin{array}{ll}{x^{2} y^{\prime}=x y-y^{2},} & {y(e)=e \quad y=\frac{x}{\ln x}} \\ {x > 1}\end{array} $$

Exact solutions \(\quad\) Find the exact solutions to the following initial value problems. a. \(y^{\prime}=1+y, \quad y(0)=1\) b. \(y^{\prime}=0.5(400-y) y, \quad y(0)=2\)

Solve the differential equations. \(x y^{\prime}-y=2 x \ln x\)

Solve the differential equation in Exercises \(9-18\). $$ 2 \sqrt{x y} \frac{d y}{d x}=1, \quad x, y>0 $$

The autonomous differential equations in Exercises \(9-12\) represent models for population growth. For each exercise, use a phase line analysis to sketch solution curves for \(P(t),\) selecting different starting values \(P(0)\) (as in Example 5 ). Which equilibria are stable, and which are unstable? $$ \frac{d P}{d t}=1-2 P $$

Use the Euler method with \(d x=0.5\) to estimate \(y(5)\) if \(y^{\prime}=y^{2} / \sqrt{x}\) and \(y(1)=-1 .\) What is the exact value of \(y(5) ?\)

Logistic differential equation Show that the solution of the differential equation $$ \frac{d P}{d t}=r(M-P) P $$ is $$ P=\frac{M}{1+A e^{-r M t}} $$ where \(A\) is an arbitrary constant.

The autonomous differential equations in Exercises \(9-12\) represent models for population growth. For each exercise, use a phase line analysis to sketch solution curves for \(P(t),\) selecting different starting values \(P(0)\) (as in Example 5 ). Which equilibria are stable, and which are unstable? $$ \frac{d P}{d t}=P(1-2 P) $$

Solve the differential equations. \(x \frac{d y}{d x}=\frac{\cos x}{x}-2 y, \quad x>0\)

Use the Euler method with \(d x=1 / 3\) to estimate \(y(2)\) if \(y^{\prime}=y-e^{2 x}\) and \(y(0)=1 .\) What is the exact value of \(y(2) ?\)

Solve the differential equation in Exercises \(9-18\). $$ \frac{d y}{d x}=x^{2} \sqrt{y}, \quad y>0 $$

Catastrophic solution Let \(k\) and \(P_{0}\) be positive constants. a. Solve the initial value problem? $$ \frac{d P}{d t}=k P^{2}, \quad P(0)=P_{0} $$ b. Show that the graph of the solution in part (a) has a vertical asymptote at a positive value of \(t .\) What is that value of \(t ?\)

The autonomous differential equations in Exercises \(9-12\) represent models for population growth. For each exercise, use a phase line analysis to sketch solution curves for \(P(t),\) selecting different starting values \(P(0)\) (as in Example 5 ). Which equilibria are stable, and which are unstable? $$ \frac{d P}{d t}=2 P(P-3) $$

Solve the differential equations. \((t-1)^{3} \frac{d s}{d t}+4(t-1)^{2} s=t+1, \quad t>1\)

Solve the differential equation in Exercises \(9-18\). $$ \frac{d y}{d x}=e^{x-y} $$

Extinct populations Consider the population model $$ \frac{d P}{d t}=r(M-P)(P-m) $$ where \(r>0, M\) is the maximum sustainable population, and \(m\) is the minimum population below which the species becomes extinct. a. Let \(m=100,\) and \(M=1200\) , and assume that \(m

The autonomous differential equations in Exercises \(9-12\) represent models for population growth. For each exercise, use a phase line analysis to sketch solution curves for \(P(t),\) selecting different starting values \(P(0)\) (as in Example 5 ). Which equilibria are stable, and which are unstable? $$ \frac{d P}{d t}=3 P(1-P)\left(P-\frac{1}{2}\right) $$

Solve the differential equations. \((t+1) \frac{d s}{d t}+2 s=3(t+1)+\frac{1}{(t+1)^{2}}, \quad t>-1\)