Question

The differential equation representing the family of hyperbolas $a^2{x}^2-b^2{y}^2=c^2$ is

• A. $\frac{y^{\prime \prime }}{y^{\prime }}+\frac{y^{\prime }}{y}=\frac{1}{x}$
• B. $\frac{y^{\prime \prime }}{y^{\prime }}+\frac{y^{\prime }}{y}=\frac{1}{x^2}$
• C. $\frac{y^{\prime \prime }}{y^{\prime }}-\frac{y^{\prime }}{y}=\frac{1}{x}$
• D. $\frac{y^{\prime \prime }}{y^{\prime }}=\frac{y}{y^{\prime }}-\frac{1}{x}$

Answer 1

Answer: (A)

Differentiating the equation twice w.r.t. $x,$ we get
$2a^{2}x-2b^{2}y{y}^{\prime }=0,a^2-b^2\left(y^{\prime 2}+y{y}^{\prime \prime }\right)=0$
Eliminating $a^2$ and $b^2,$ we get the differential equation
$\frac{y^{\prime \prime }}{y^{\prime }}+\frac{y^{\prime }}{y}=\frac{1}{x}$