Question

Form the differential equation of the family of hyperbolas having foci on $x$-axis and centre at origin

Answer 1

Equation of a hyperbola is given by $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Differentiating both the sides, we get

$\frac{2x}{a^2}-\frac{{2yy}^{{}^{\prime }}}{b^2}=0$...........(1)

Again differentiating, we get

$\frac{1}{a^2}-\left(\frac{1}{b^2}\right)\left[\right(y^{{}^{\prime }}{)}^2+y{y}^{\prime \prime }]$

$\frac{1}{a^2}=\left(\frac{1}{b^2}\right)\left[\right(y^{{}^{\prime }}{)}^2+y{y}^{\prime \prime }]$

Putting this in equation (1), we get

$\left(\frac{x}{b^2}\right)\left[\right(y^{{}^{\prime }}{)}^2+y{y}^{\prime \prime }]-\frac{{yy}^{{}^{\prime }}}{b^2}=0$

Multiply by $b^2$ and after simplification, we get

$xy{y}^{\prime \prime }+x(y^{{}^{\prime }}{)}^2-{yy}^{{}^{\prime }}=0$