Problem

The direction field for is shown in dy/dx = 4x/y Figure 1.13.

(a) Verify that the straight lines y = ± 2x are solution curves, provided x ≠ 0.

(b) Sketch the solution curve with initial condition y (0) = 2.

(c) Sketch the solution curve with initial condition y(2) = 1.

(d) What can you say about the behavior of the above solutions as x + ∞? How about - ∞

Step 1 ;

In this problem given that

Consider

Given that the straight lines

is verified.

Then the below graph shows the solution equation .

Step 2 :

(b)

Solve for

.’.

Using the initial condition

The direction formula of the graph is

Step 3 :

(c)

Using the initial condition

.’.

The graph is

Step 5 :

(d)

Below graph shows the and

Hence the result.