. . Formation of Differential Equations from Geometrical Problems Given a family of functions parameterized by some constants, a differential equation can be formed by eliminating those constants of this family. For instance, the elimination of constants A and B from y A B , yields a differential equation d y Consider an equation of a family of curves, which contains n arbitrary constants.
To form a differential equation not containing any of these constants, let us proceed as follows: Differentiate the given equation successively n times, getting n differential equations. Then eliminate n arbitrary constants from ( n + equations made up of the given equation and n newly obtained equations arising from n successive differentiations. The result of elimination gives the required differential equation which must contain a derivative of the n th order. Example .
Find the differential equation for the family of all straight lines passing through the origin. The family of straight lines passing through the origin is y mx , where m is an arbitrary constant. … ( ) Differentiating both sides with respect to x , we get … ( ) From ( ) and ( ), we get y x dy . This is the required differential equation.
Observe that the given equation y mx contains only one arbitrary constant and thus we get the differential equation of order one. Example . Form the differential equation by eliminating the arbitrary constants A and B from A B Given that y = A B ... ( ) Differentiating ( ) twice successively, we get dx = − A B x .
... ( ) d y = − = − A B A B sin ) x . ... ( ) Fig.
. = O = − = − - - Ordinary Differential Equations Substituting ( ) in ( ), we get d y as the required differential equation. Example . Find the differential equation of the family of circles passing through the points ( , ) a and − a , .
A circle passing through the points a , ) and