The general solution of the differential equation
` (dy)/(dx) = ((1+y^(2)))/(xy(1+x^(2)))` is

Options

A. `(1+x^(2))(1+y^(2))=C`
B. `(1+x^(2))(1+y^(2))=Cx^(2)`
C. `(1-x^(2))(1-y^(2))=C`
D. `(1+x^(2))(1+y^(2))=Cy^(2)`

Answer

b

Explanation

Given differential equation can be rewritten as
` y/(1+y^(2))dy= (dx)/(x(1+x^(2)))`
`rArr 1/2 int (2y)/(1+y^(2))dy = (dx)/(x(1+x^(2)))`
` rArr 1/2 int (2y)/((1+y^(2)))dy=1/2 int (dt)/(t(t+1))`
[ put `x^(2)` = t and 2x dx = dt in RHS integral ]
` rArr 1/2 int ((2y" "dy))/(1+y^(2))=1/2 int (1/t - 1/(1+t))dt`
` rArr 1/2 * log (1+y^(2)) = 1/2 [ log t - log (1+t)] +1/2 log C`
` rArr log 1+y^(2) = logx^(2)- log (1+x^(2))+log C`
` rArr log (1+y^(2))(1+x^(2))= log Cx^(2)`
` rArr (1+y^(2))(1+x^(2))= Cx^(2)`