Question

Match the elements of list I, which have differential equations, with elements of list II, which have solutions of differential equations:

List I	List II
A) $y{y}_1={\sec }^{2}x$	1) $y{\sec }^{2}x=\sec x+c$
B) $y_1=x\sec y$	2) $xy=\cos y+c$
C) $y_1+\left(2y\tan x\right)=\sin x$	3) $xy=\sin x+c$
D) $x{y}_1+y=\cos x$	4) $y^2=2\tan x+c$
	5) $x^2=2\sin y+c$

Then, the correct order for $ABCD$ is:

• A. $A-3,B-2,C-5,D-4$
• B. $A-4,B-1,C-2,D-3$
• C. $A-4,B-5,C-1,D-3$
• D. $A-3,B-5,C-1,D-2$

Answer 1

Answer: (C)

$A-4,B-5,C-1,D-3$

We will differentiate the elements of the list II, which contains the solutions to the differential equations, to get the differential equations
1)$y{\sec }^{2}x=\sec x+c$

Differentiating both sides w.r.t. $x$,

$y\left(2{\sec }^{2}x\tan x\right)+y_1{\sec }^{2}x=\sec x\tan x$
$2y\tan x+y_1=\sin x$ $\to$ C
2)$xy=\cos y+c$

Differentiating both sides w.r.t. $x$,
$x{y}_1+y=-\sin y{y}_1$
3)$xy=\sin x+c$

Differentiating both sides w.r.t. $x$,
$x{y}_1+y=\cos x$ $\to$ D
4)$y^2=2\tan x+c$

Differentiating both sides w.r.t. $x$,

$2y{y}_1=2{\sec }^{2}x$
$y{y}_1={\sec }^{2}x$ $\to$ A
5)$x^2=2\sin y+c$

Differentiating both sides w.r.t. $x$,
$2x=2\cos y{y}_1$
$x=y_1\cos y$ $\to$ B

The matching order for $ABCD$ is $4513.$

Hence, option $C.$