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Question
Find the particular solution of the differential equation `e^xsqrt(1-y^2)dx+y/xdy=0` , given that y=1 when x=0
Solution
We have:
`e^xsqrt(1−y2)dx+y/x dy=0 `
`e^xsqrt(1−y2)dx=-y/x dy..........(1)`
Separating the variables in equation (1), we get:
`xe^xdx=-y/sqrt(1-y^2)dy.........(2)`
Integrating both sides of equation (2), we have:
`int xe^xdx=-inty/sqrt(1-y^2)dy ............(3)`
`Now,intxe^xdx=xe^x-e^x+C_1=e^x(x-1)+C_1.......(4)`
`"Let " I=-inty/sqrt(1-y^2)dy`
putting `1-y^2=t` we get,
`-2ydy=dt`
`-ydy=dt/2`
`I=1/2intdt/sqrtt`
`=1/2xx2t^(1/2)+C_2`
`=t^(1/2)+C_2`
`=(1-y^2)^(1/2)+C2.......(5)`
Putting the values in equation (3), we get
`e^x(x-1)+C_1=(1-y^2)^(1/2)+C_2`
`e^x(x-1)=(1-y^2)^(1/2)+C, "where " C=C_2-C_1.......(6)`
on putting y=1 and x=0 in equation (6) we get C=-1
The particular solution of the given differential equation is `e^x(x-1)=(1-y^2)-1`
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