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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 $2xt = y + t^2$
On differentiating, we get $2t = y_1$
On putting this value in the above equation, we get
$xy_{1}=y+\left(\frac{y_{1}}{2}\right)^{2}$
$\Rightarrow 4xy_{1}=4y+y^{2}_{1}$
The order of this equation is $1$.