Question

The differential equation of the family of curves, $x^2=4b(y+b),b\in R$, is:

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

Answer 1

The correct option is B $x(y^{\prime }{)}^2=x+2y{y}^{\prime }$

$x^2=4b(y+b)$
Differentiating both the sides w.r.t. $x$, we get
$\Rightarrow 2x=4b{y}^{\prime }$
$\Rightarrow b=\frac{x}{2y^{\prime }}$
Putting the value of $b$ in $\left(1\right),$ we get
$\Rightarrow x^2=\frac{2x}{y^{\prime }}(y+\frac{x}{2y^{\prime }})$

$\Rightarrow x^2=\frac{2xy}{y^{\prime }}+\frac{x^2}{y^{\prime 2}}$

$\Rightarrow x(y^{\prime }{)}^2=2y{y}^{\prime }+x$