Match the following

	Column – I		Column – II
(A)	If the function $y=e^{4x}+2e^{-x}$ is a solution of differential equation $\frac{\frac{d^{3}y}{d{x}^3}-13\frac{dy}{dx}}{y}=k$ then the value of $k/3$ is	(p)	3
(B)	Number of straight lines which satisfy the differential equation $\frac{dy}{dx}+x{\left(\frac{dy}{dx}\right)}^2-y=0$ is	(q)	4
(C)	If real value of $m$ for which the substitution $y=u^m$ will transform the differential equation $2x^{4}y\frac{dy}{dx}+y^4=4x^6$ into homogeneous then the value of $2m$ is	(r)	2
(D)	If the solution of differential equation $x^2\frac{d^{2}y}{d{x}^2}+2x\frac{dy}{dx}=12y$ is $y=A{x}^m+B{x}^{-n}$  then $|m+n|$ is	(s)	1

A)        $y=e^{4x}+2e^{-x};$ Find $y\text{'},y",y\text{'}"$ 

                        Consider $\frac{\frac{d^{3}y}{d{x}^3}-13\frac{dy}{dx}}{y}=12;$

                                    $K=12;\frac{K}{3}=\frac{12}{3}=4$

B)        Let $y=mx+c;\frac{dy}{dx}+x{\left(\frac{dy}{dx}\right)}^2-y=0$;

            $m+x{m}^2=mx+c;m=c$         

comparing like terms $m=m^2;m=0,1$

C)        $y=u^m;\frac{dy}{dx}=m{u}^{m-1}\frac{du}{dx};$

            $2x^{4}y\frac{dy}{dx}+y^4=4x^6;\frac{du}{dx}=\frac{4x^6-u^{4m}}{2x^{4}m{u}^{2m-1}}$

            Homogeneous $4m=6;m=6/4=3/2;2m=3$

D)        $y=A{x}^m+B{x}^{-n};y=Am{x}^{m-1}-nB{x}^{-n-1}$

            $y_2=Am(m-1)x^{m-2}+n(n+1)B{x}^{-n-2}$

            $\Rightarrow x^2{y}_2+2x{y}_1=12y$

            $Am(m-1)x^m+n(n+1)B{x}^{-n}+2Am{x}^m-2n{x}^{-n}$

            $=12(A{x}^m+B{x}^{-n});m^2-m-12=0;m=-4,3;$

            $n(n+1)B-2nB=12B$

            $n^2-n-12=0;n=4,-3;|m+n|=|-4+3|=1$