### 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

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

Now if in (1) we replace

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

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**m**. 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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