Question

Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point.

y' = y - 2x,        (1, 0)

Answer 1

Step 1:

The differential equation is and point is .

Slope field is

A direction field is graphical representation of the solutions of a first order differential equation.

Create a table to compute the slope at several values of and .

	-3	-2	-1	0	1	2	3
	0	1	2	3	0	-1	2
	-6	-3	0	3	-2	-5	-4

Now draw the short line segments with their slopes at respective points.

The result is the direction field of the differential equation.

Graph the directional field of differential equation:

Comments

Step 2:

Observe the table:

The slope of the differential equation at point is .

Now draw a solution curve so that it move parallel to the near by segments.

The resulting curve is solution curve which passes through .

Note :

The curve in pink color is the solution curve passing through the point .

Solution:

Directional field of differential equation is 

Solution curve passing through is