Question:

Consider the differential equation dy/dx = x^2 - 3x - 2.

1. Sketch the slope field for this differential equation on the axes provided below.

2. Determine the particular solution of the differential equation that passes through the point (1, 2), and sketch the solution curve on the same set of axes from part 1.

3. Find the equation of any horizontal asymptotes, if they exist, for the solution curve obtained in part 2.

Solution:

1. To sketch the slope field, we will calculate the slope at various points and draw short line segments representing the slope at each point. We will plug in values of x and y into the given differential equation and calculate dy/dx. The sign of dy/dx will determine whether the line segment should point upward or downward.

   Let's start by calculating the slope at various points. We will use a step size of 1 for ease of calculation:

   x	y	dy/dx
   -2	-2	-2
   -1	-1	0
   0	0	-2
   1	1	-2
   2	2	2
   3	3	8
   4	4	14

   Now, we can sketch the slope field on the given axes:

   

2. To determine the particular solution that passes through the point (1, 2), we will use the initial value problem (IVP) approach. We integrate the given differential equation and solve for the constant of integration using the initial condition.

   Integrating the given differential equation, we have:

   ∫ dy = ∫ (x^2 - 3x - 2) dx

   y = (x^3)/3 - (3x^2)/2 - 2x + C

   Now, we substitute the coordinates of the point (1, 2) to find the value of the constant C:

   2 = (1^3)/3 - (3(1)^2)/2 - 2(1) + C

   2 = 1/3 - 3/2 - 2 + C

   2 = -19/6 + C

   C = 2 + 19/6

   C = 31/6

   Therefore, the particular solution is given by:

   y = (x^3)/3 - (3x^2)/2 - 2x + 31/6

   We can plot this solution curve on the same set of axes as the slope field:

   

3. To determine the horizontal asymptotes, we observe the behavior of the solution curve as x approaches positive or negative infinity. If the solution curve approaches a constant value as x tends to infinity, then there exists a horizontal asymptote.

   Taking the limit of the solution equation as x approaches infinity, we have:

   lim(x->∞) [(x^3)/3 - (3x^2)/2 - 2x + 31/6]

   As x approaches infinity, the terms involving x^3 and x^2 will dominate. The constant terms and the higher degree terms become negligible.

   lim(x->∞) [(x^3)/3 - (3x^2)/2 - 2x + 31/6] ≈ lim(x->∞) [(x^3)/3 - (3x^2)/2]

   Applying L'Hôpital's rule, we can find the limit of the ratio of the leading coefficients:

   lim(x->∞) [(x^3)/3 - (3x^2)/2] ≈ lim(x->∞) (3x^2 - 3x) / (3x - 2)

   Now, divide every term by x^2:

   lim(x->∞) (3 - 3/x) / (3/x - 2/x^2)

   lim(x->∞) (3 - 3/x) / (3/x - 2/x^2) = 3/0

   As x approaches infinity, the limit of the function is undefined.

   Therefore, there are no horizontal asymptotes for the solution curve.

   

   Note: The solution curve shown above on the graph may not be an accurate representation. It is only intended to illustrate the general behavior of the solution.