Find the differential equation that will represent the family of curves given by (a, b: parameter):
x² + y² = a x³

Solution

The given differential equation is

straight x squared plus straight y squared space equals space ax cubed

...(1)

2 straight x plus 2 straight y dy over dx space equals 3 ax squared

...(2)
Multiply (1) by 3 and (2) by x, we get,

3 straight x squared plus 3 straight y squared space equals space 3 ax cubed

...(3)

2 straight x squared plus 2 xy dy over dx space equals space 3 ax cubed

...(4)
Subtracting (4) from (3), we get

straight x cubed plus 3 straight y squared minus 2 xy dy over dx equals 0

straight x squared plus 3 straight y squared equals 2 xy dy over dx

which is required differential equation.

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