Question

differential equation of

Answer 1

\[(y-b)^{2}=4(x-a)\]

Answer 2

diff. twice w.r.t. x
\[2\left( y-b \right)y \prime=4 ...(1)\]
\[2\left( y-b \right)y \prime \prime +2y \prime y \prime=0\]
\[\left( y-b \right)y \prime \prime+\left( y \prime \right)^2=0...(2)\]
\[from (1)~ y-b=\frac{ 2 }{ y \prime }\]
put in (2)
\[\frac{2 y \prime \prime }{ y \prime }+\left( y \prime \right)^2=0\]
\[2y \prime \prime+ \left( y \prime \right)^3=0\]