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Q.
The order of the differential equation of all tangent lines to the parabola $y = x ^{2}$, is
Differential Equations
Solution:
The parametric form of the given equation is
$x = t , y = t ^{2}$
The equation of any tangent at $t$ is
$2 xt = y + t ^{2}$
On differentiating, we get
$2 t = y _{1}$.
On putting this value in the above equation, we get
$xy _{1}= y +\left(\frac{ y _{1}}{2}\right)^{2}$
$ \Rightarrow 4 xy _{1}=4 y + y _{1}^{2}$
The order of this equation is 1 .