Find the particular solution of the differential equation x (1 + y^{2}) dx – y (1 + x^{2}) dy = 0, given that y = 1 when x = 0.

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#### 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)`

Concept: General and Particular Solutions of a Differential Equation

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