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Question
Find the particular solution of the differential equation x (1 + y2) dx – y (1 + x2) dy = 0, given that y = 1 when x = 0.
Solution
Given equation can be written as
`x/(1+x^2)dx-y/(1+y^2)dy=0`
Integrating to get
`1/2 log (1+x^2)-1/2log(1+y^2)=logc_1`
`=>log(1+x^2)-log(1+y^2)=logc_1^2=logc`
`therefore (1+x^2)/(1+y^2)=c`
`x=0,y=1=>c=1/2`
`therefore 1+y^2=2(1+x^2) or y=sqrt(2x^2+1)`
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