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Solve the differential equation x y(dy)/...

Solve the differential equation `x y(dy)/( dx)=(1+y^2)/(1+x^2)(1+x+x^2)`

Text Solution

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We have `sqrt((1+x)^(2)(1+y)^(2))=-xy(dy)/(dx)`
`therefore sqrt(1+x^(2))/(x) dx =-y/sqrt(1+y^(2))dy`
Integrating both sides,
`int(1+x^(2))/(xsqrt(1+x^(2)))dx=-int(y)/sqrt(1+y^(2))dy`
`rArr int (dz)/(z^(2)-1)+sqrt(1+x^(2))=-sqrt(1-y^(2))+C`
`rArr 1/2log_(e)|(z+1)/(z-1)|+sqrt(1+x^(2))=-sqrt(1+y^(2))+C`
`rArr 1/2log_(e)|(sqrt(1+x^(2)-1))/(sqrt(1+x^(2))+1)|+sqrt(1+x^(2))=-sqrt(1+y^(2))+C`
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