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Question

A solution of differential equation is a relation between the variables (independent and dependent), which is free of derivatives of any order and satisfies the differential equation identically.
Differential equation General solution (A) 2xy2dx=ex(dyydx) (P) 4x3y+6ex3y3=cy(B) y(2x2y+ex)dx=(ex+y3)dy(Q) x+loge(x2+y2)=c(C) (x2+y2+2x)dx+2ydy=0(R) x2loge(x2y2)+y3=cx2(D) (2x2y2y4)dx+(2x3+3xy3)dy=0(S) x2y+ex=cy
Here, c is an integration constant.

Which of the following is the CORRECT combination?

A
(D)(Q)
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B
(A)(R)
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C
(B)(P)
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D
(C)(S)
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Solution

The correct option is C (B)(P)
(A)

2xy2dx=ex(dyydx)2xexy2=dydxy1y2dydx1y=2xex
Taking 1y=tdtdx=1y2dydx
dtdx+t=2xex

I.F.=edx=ex
tex=2x dx
exy=x2+c1
x2y+ex=cy, where c=c1
(A)(S)

(B)

y(2x2y+ex)dx=(ex+y3)dy2x2y2dxy3dy+yexdxexdy=0(2x2dxydy)+ex(ydxdy)y2=0
(2x2dxydy)+d(exy)=0
On integration, we get
2x33y22+exy=c2
4x3y+6ex3y3=cy, where c=6c2
(B)(P)

(C)

(x2+y2+2x)dx+2ydy=0x2+y2+2x+2ydydx=0(x2+y2)+ddx(x2+y2)=0d(x2+y2)x2+y2=dx
On integration, we get
loge(x2+y2)=x+cx+loge(x2+y2)=c
(C)(Q)

(D)

(2x2y2y4)dx+(2x3+3xy3)dy=02x2y2y4+2x3dydx+3xy3dydx=02x2(y+xdydx)+y3(3xdydx2y)=02(y+xdydx)+y(3y2xdydx2y3x2)=02(1+xydydx)+(3y2xdydx2y3x2)=02x+2ydydx+(3y2x2dydx2y3x3)=02x+2ydydx+ddx(y3x2)=02dxx+2dyy+d(y3x2)=0
On integration, we get
lnx2+lny2+y3x2=cx2loge(x2y2)+y3=cx2
(D)(R)

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