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.

$$\begin{gathered} \mathit{y}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{=}\mathbf{-}\mathit{x} \\ \mathit{a}\mathbf{)}\mathit{y}\left(1\right)\mathbf{=}\mathbf{1} \\ \mathit{b}\mathbf{)}\mathit{y}\left(0\right)\mathbf{=}\mathbf{4} \end{gathered}$$

Short answer

Answer:

a). The curve is passing through the point$\left(1,1\right)$.

b). The curve is passing through the point$\left(0,4\right)$.

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 $\left(x,y\right)$of the grid with a slope $f\left(x,y\right)$, then the collection of all these line elements is called a direction field or a slope field of the differential equation$\frac{dy}{dx}=f\left(x,y\right)$.

Sketch the graph.

The given differential equation can be written as,

$\frac{dy}{dx}=-\frac{x}{y}$

The directional field plot is shown below,

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

Note that the implicit solution is a complete circle. However, since we are given an initial condition, the solution curve is only the semicircle on which the initial condition lies. We stop sketching at the points when $y=0$because the differential equation is undefined for that value.

a). The curve is passing through the point$\left(1,1\right)$.

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

b). The curve is passing through the point$\left(0,4\right)$.

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