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 .

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