Question

In Problems 5–12 use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points.

Short answer

a). The curve is passing through the point\((0,1)\).

b). The curve is passing through the point\(( - 2, - 1)\).

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.

The given differential equation is,

\(f(x,y) = \frac{1}{y}\)

The directional field plot is shown below,

For all \(x\), at\(y = 0\) , the slope \(f(x,y),\) is undefined.

For a fixed value of \(x\), the slope \(f(x,y) > 0\) for \(y > 0\) and \(f(x,y) < 0\) for \(y < 0\).

For a fixed value of \(x\) below the \(x\)-axis, the linear elements have a negative slope and almost become horizontal as \(y \to - \infty ,x \to \infty \) , and similarly, for a fixed value of\(x\) below the \(x\)-axis, the linear elements have a positive slope and almost becomes horizontal as \(y \to \infty ,x \to \infty \)

a).The curve is passing through the point \((0,1)\).

From the above directional field, their plot color is red.

b).The curve is passing through the point \(( - 2, - 1)\).

From the above directional field, their plot color is blue.