− a , has its centre on y - axis. Let , b ) be the centre of the circle. So, the radius of the circle is Therefore the equation of the family of circles passing through the points a , ) and − a , is ) = is an arbitrary constant. ...
( ) Differentiating both sides of ( ) with respect to x , we get b dy = ⇒ = − ⇒ y . Substituting the value of b in equation ( ), we get = a ⇒ + = a y dy + ⇒ = , which is the required differential equation. Example . Find the differential equation of the family of parabolas y ax , where a is an arbitrary constant.
The equation of the family of parabolas is given by y ax , a is an arbitrary constant. ... ( ) Differentiating both sides of ( ) with respect to x , we get y dy y dy ⇒ Substituting the value of a in ( ) and simplifying, we get dy = as the required differential equation. Example .
Find the differential equation of the family of all ellipses having foci on the x -axis and centre at the origin. The equation of the family of all ellipses having foci on the x -axis and centre at the origin is given by x > ... ( ) where a and b are arbitrary constants. Differentiating equation ( ) with respect to x , we get = ⇒ ...
( ) Differentiating equation ( ) with respect to x , we get y d y + = ⇒ = − + y d y Substituting the value of a in equation ( ) and simplifying, we get + y d y dx = ⇒ +