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Question

If the transformation z=logtan(x2) reduces the differential equation d2ydx2+cosxdydx+4y cosec2x=0 into d2ydx2+Ay=0 then the value of A is

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Solution

We have
dzdx=12sec2(x/2)tan(x/2)=1sinx=cosecx
dydx=dydz.dzdx=cosecxdydz
d2ydx2=ddx(dydx)=ddz(dydx)dzdx
=ddz(cosecxdydz)cosecx
=(cosecxd2ydx2cosecxcotxdxdzdydz)cosecx
=cosec2xd2ydz2cosecxcotxdydz
Putting these values in the given differential equation, we have
d2ydx2+cotxdydx+4ycosec2x=0
cosec2xd2ydx2cosecxcotxdydz+cotxcosecxdydz+4ycosec2x=0
cosec2x(d2ydz2+4y)=0 d2ydz2+4y=0
A=4

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