Answer to Question #135846 in Differential Equations for Nikhil Singh

Question

The differential equation of a family of tangent lines to the parabola y=x^2, given by 2xt=y+t, t being a parameter,4(y-xdy/dx)+(dy/dx)^2=0.
True or false with statement

Answer 1

The parametric form of the given equation is $x=t, y=t^2.$

The equation of any tangent at $t$  is $2xt=y+t^2.$

Differentiating, we get $2t=dy/dx$

Putting this value in the equation of tangent, we have 

$$x(dy/dx)=y+1/4(dy/dx)^2$$

$$4(y-x(dy/dx))+(dy/dx)^2=0$$

True.