Question

Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. To print an enlarged copy f the graph, go to the website www.mathgraphs.com.

Answer 1

Step 1:

The differential equation 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 .

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

The result is the slope field of the differential equation.

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

In the similar manner, draw few more solutions of the differential equation.

Graph the slope filed of differential equation and their solutions:

Note : The curve in pink color are the solution curves.

Step 2:

Consider .

Re-write the equation.

Integrate on each side.

Take exponents of each side.

Properties of natural logarithms: .

The solution of the differential equation is .

Solutions:

Graph the slope filed of differential equation and their solutions is

The solution of the differential equation is .