Match the following family of curve with their differential equations

	Column – I		Column – II
A	$y=cx+c^2$	P	$y\left(1-{\left(\frac{dy}{dx}\right)}^2\right)=2x\frac{dy}{dx}$
B	$y=a{e}^{2x}+b{e}^{3x}$	Q	$x\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}-xy+x^2-2=0$
C	$y^2=4a\left(x+a\right)$	R	$\frac{d^{2}y}{d{x}^2}-5\frac{dy}{dx}+6y=0$
D	$xy=a{e}^x+b{e}^{-x}+x^2$	S	$y=x\frac{dy}{dx}+{\left(\frac{dy}{dx}\right)}^2$

A) $\frac{dy}{dx}=c$  , on putting c in equation we get $y=x\frac{dy}{dx}+{\left(\frac{dy}{dx}\right)}^2$  
B)  Different two & elements a & b 
C)  $2y\frac{dy}{dx}=ya$ substitute a in equation
D) Multiplying equation by ${\text{e}}^{\text{x}}$ , we get  $x{y}^{ex}-a{e}^{2x}+b+x^2{e}^x$
 on differentiating, we get  $y{e}^x+xy,ex+xy{e}^x=2x{e}^{2x}+x{e}^x+2x{e}^x.$ 
Divide by ${\text{e}}^{\text{2x}}$   &    differentiates agree