Question

From x² + y² + 2ax + 2by + c = 0, derive a differential equation not containing a, b and c.

Answer 1

We have,

x² + y² + 2ax + 2by + c = 0         .....(i)

Differentiating (i) with respect to x, we get

\[2x + 2yy' + 2a + 2by' = 0\]
Again differentiating with respect to `x`, we get
\[2 + 2 \left( y' \right)^2 + 2yy'' + 2by'' = 0\]
\[1 + \left( y' \right)^2 + yy'' + by'' = 0\]
\[b = \frac{- \left( 1 + \left( y' \right)^2 + yy" \right)}{y ''}\]
We have,
\[1 + \left( y' \right)^2 + yy'' + by'' = 0\]
Again differentiating with respect to `x`, we get
\[2y'y'' + y'y '' + yy''' + by''' = 0\]
On substituting the value of `b` we get,
\[3y'y'' + yy''' + \left( \frac{- \left( 1 + \left( y' \right)^2 + yy " \right)}{y''} \right)y''' = 0\]
\[3y' \left( y'' \right)^2 + yy '' y''' - y''' - \left( y' \right)^2 y''' - yy'''y " = 0\]
\[3y' \left( y " \right)^2 = y'''\left( 1 + \left( y' \right)^2 \right)\]