Question

Consider the family of curves $A{x}^2+B{y}^2=1$. If this family of curve is represented by the differential equation $P.y^{\prime \prime }+Q(y^{\prime }{)}^2=y{y}^{\prime }$, where P and Q are functions of x and y, then which of the following statement (s) is/are correct? $(y^{\prime }=\frac{dy}{dx}\text{and}{y}^{\prime \prime }=\frac{d^{2}y}{d{x}^2})$

• A. $P=xy$
• B. $Q=xy$
• C. $P+Q=x(y+1)$
• D. $P-Q=y(x-1)$

Answer 1

Answer: (A), (C)

The correct options are
A $P=xy$
C $P+Q=x(y+1)$

$A=\frac{1-B{y}^2}{x^2}$
Differentiating
$0=x^2(-2By{y}^{\prime })-(1-B{y}^2).2x$
$B=\frac{1}{y^2-xy{y}^{\prime }}$
$\frac{1}{B}=y^2-xy{y}^{\prime }\Rightarrow 2y{y}^{\prime }-xy{y}^{\prime \prime }-y{y}^{\prime }-x(y^{\prime }{)}^2=0$
$\Rightarrow xy{y}^{\prime \prime }+x(y^{\prime }{)}^2=y{y}^{\prime }$
So, $P=xy,Q=x$