Thursday, 12 December 2013

Method of Isoclines

It is only in the limited number of cases that a differential equation may be solve analytically be the preceding methods , and in many practical cases where the solution of a differential equation is needed under given initial conditions and the above methods  fail , a graphical method , the method of isoclines is sometimes adopted . We proceed to explain below this method in case of simple differential equation of the first order .

Let us consider an equation of the type

dy/dx = f (x ,y ) ------------(1)

As already explained before , the general solution of this equation involves one arbitrary constant of integration . and hence represents a family of curves and in general , one member of the family passes through a given point  ( x , y ) .

Now if in  (1)  we replace   dy/dx  by   m   we get an equation   f( x , y) = m    , which for any particular numerical value of   m  represents a curve , at every point of which the value of  dy/dx  i.e.  the slope of the tangent line to the family of curves represented by the general solution of   (1)  is the same as the numerical value of   . This curve   f(x , y) = m   is called an Isoclinal or Isocline .

Which may be graphically constructed on a graph paper


Through different points on anyone isocline , short parallel lines are drawn having their common slope equal to the particular value of  m  for that Isocline .

No comments:

Post a Comment

Our Latest Post

How to Evaluate an Integral

In this video you will see how to evaluate an Integral. This video shows an example, by this example you will see about how to evaluate an ...

Popular Post