I am new to differential equations any help on this problem

   I am new to differential equations any help on this problem would be appreciated. even just some advise on where to start. thanks

Solution

Alert: The image is very smudgy, cant make out the letters.

Hope that the following helps you:

A differential equation (simplest type) is a relation of the form

                         dy/dx=f(x,y) with y(x[0])=y[0]...................(1)

For example,

                        dy/dx= sin (x) with y(0) = 0..................(2)

                        dy/dx= sin (x²) with y(0) =0 ..................(3)

Compare (2) and (3).

It is easily seen that y(x)= -cos (x)+1 is a(in fact the unique) solution of (2).

The equation (3) too has a unique solution, but it cannot be solved explicitly, because there is no standard function g(x) such that g\'(x) =sin (x²). Nevertheless, a solution exists and it could be found numerically or geometrically.

The material posted in the question concerns the geometric aspect.

As dy/dx represents the slope of the tangent to the curve y=y(x) , we can always draw the tangent field because f(x,y) can be calculated for every point (x,y) in the plane.

Using this picture, one can understand the behaviour of the solutions , without actually solving the equation explicitly.

The following link (in fact the entire material on Differential equations is a very good, easy introduction to these aspects)

http://tutorial.math.lamar.edu/Classes/DE/PhasePlane.aspx

Hope this helps you . Please repost a better image of the problem