In this exercise, you will need to solve the differential equation given below.

$$x(x-1) d y-y d x=0$$

The autonomous differential equations 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). Which equilibria are stable, and which are unstable?

$$\frac { d P } { d t } = P ( 1 - 2 P )$$

Why is the exponential model unrealistic for predicting long-term population growth? How does the logistic model correct for the deficiency in the exponential model for population growth? What is the logistic differential equation? What is the form of its solution? Describe the graph of the logistic solution.

In this exercise, you will need to solve the differential equation given below.

$$y^{\prime}=x e^{x-y} \csc y$$

Answer the exercise given below. Write the correct answer.

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 the exercise given below. Write your answer.

$$\frac{d y}{d x}=x^2 \sqrt{y}, \quad y>0$$

Solve the differential equation.

$$x \frac { d y } { d x } = \frac { \cos x } { x } - 2 y , \quad x > 0$$

Under some conditions, the result of the movement of a dissolved substance across a cell’s membrane is described by the equation

$$\frac { d y } { d t } = k \frac { A } { V } ( c - y ).$$

In this equation, y is the concentration of the substance inside the cell and dy/dt is the rate at which y changes over time. The letters k, A, V, and c stand for constants, k being the permeability coefficient (a property of the membrane), A the surface area of the membrane, V the cell’s volume, and c the concentration of the substance outside the cell. The equation says that the rate at which the concentration changes within the cell is proportional to the difference between it and the outside concentration. a. Solve the equation for y(t), using

$$y_0$$

to denote y(0). b. Find the steady-state concentration,

$$\lim _ { t \rightarrow \infty } y ( t ).$$

Solve the differential equation.

$$y ^ { \prime } = x e ^ { y } \sqrt { x - 2 }$$

Solve the differential equation exercise given below. Write the correct answer.

$$\frac{d y}{d x}=\sqrt{y} \cos ^2 \sqrt{y}$$

a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of y′ and y″. c. Sketch several solution curves.

$$\frac { d y } { d x } = ( y + 2 ) ( y - 3 )$$

Solve the differential equation.

$$x \frac { d y } { d x } + y = e ^ { x } , x > 0$$

Coasting bicycle A 66-kg cyclist on a 7-kg bicycle starts coasting on level ground at 9 m/sec. The $k$ in the equatiom is about 3.9 kg/sec.

a. About how far will the cyclist coast before reaching a complete stop? b. How long will it take the cyclist's speed to drop to 1 m/sec?

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.

$$y^{\prime}=1-\frac{y}{x}, \quad y(2)=-1, d x=0.5$$