Question

In Problems 1–4 reproduce the given computer-generated direction field. Then sketch, by hand, an approximate solution curve that passes through each of the indicated points. Use different colored pencils for each solution curve.

FIGURE 2.1.13 Direction field for Problem 2

Short answer

Answer:

We sketch the curve of approximate solutions for each of the four cases using the given points and the flow pattern of the field.

Step-by-step solution

Direction field.

If we systematically evaluate\(f\)over a rectangular grid of points in the\(xy\)-plane and draw a line element at each point\((x,y)\)of the grid with a slope\(f(x,y),\)then the collection of all these line elements is called adirection field or a slope field of the differential equation \(\frac{{dy}}{{dx}} = f(x,y)\).

Sketch the graph.

Given that,

\(\begin{aligned}{l}\frac{{dy}}{{dx}} &= 1 - xy\\a)y(0) &= 0\\b)y( - 1) &= 0\\c)y(2) &= 2\\d)y(0) &= - 4\end{aligned}\)

We are given the equation and four points through which a solution curve passes.

We sketch the curve of approximate solutions for each of the four cases using the given points and the flow pattern of the field. The graph is shown below.