Question

Form the differential equation of the family of curves represented by the equation(a being the parameter).

$(x-a{)}^2+2y^2=a^2$.

Answer 1

The equation of the one parameter family of curves is

$(x-a{)}^2+2y^2=a^2$ ………(i)

Differentiating with respect to x, we get

$2(x-a)+4y\frac{dy}{dx}=0\Rightarrow x-a=-2y\frac{dy}{dx}\Rightarrow a=x+2y\frac{dy}{dx}$

Substituting the value of a in (ii), we get

$4y^2{\left(\frac{dy}{dx}\right)}^2+2y^2={(x+2y\frac{dy}{dx})}^2\Rightarrow 2y^2-x^2=4xy\frac{dy}{dx}$

This is the required differential equation.