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Question

Consider the following differential equation:

x(ydx+xdy)cosyx=y(xdx−ydx)sinyx

Which of the following is the solution of the above equation ( c is an arbitrary constant) ?

A
xycosyx=c
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B
xysinyx=c
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C
xycosyx=c
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D
xysinyx=c
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Solution

The correct option is C xycosyx=c
x(ydx + xdy)cosyx=y(xdyydx)sinyx

ydx+xdyxdyydx=yxtanyx

Let y = vx

dy = v dx + xdv

vxdx+vxdx+x2dyvxdx+x2dyvxdx=v tanv

xdv+2vdxxdv=v tanv

1 + 2vxdxdv=v tanv

2vxdxdv=v tanv1

2dxx=(tanv1v)dv

Integrating both sides

2 log x = log |secv| - log v + log c

x2=c secvv

x2.yx=c sec(y/x)

xycosyx=c

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