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.

$$\begin{gathered} \mathit{y}\mathbf{\text{'}}\mathbf{=}\mathit{x} \\ \mathit{a}\mathbf{)}\mathit{y}\left(0\right)\mathbf{=}\mathbf{0} \\ \mathit{b}\mathbf{)}\mathit{y}\left(0\right)\mathbf{=}\mathbf{-}\mathbf{3} \end{gathered}$$

Short answer

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

b).The curve is passing through the point\((0, - 3)\).

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 directional field plot is shown below,

To draw the solution curves, follow the slope lines starting from the initial condition.

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

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

b).The curve is passing through the point\((0, - 3)\).

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