Question

solution of the differential equation `xdy-ydx=sqrt(x^2+y^2 )dx` is
A. ` y + sqrt(x^(2)+y^(2))=Cx`
B. ` y + sqrt(x^(2)+y^(2))=Cx^(2)`
C. ` y + sqrt(x^(2)+y^(2))=C`
D. None of these

Answer 1

Correct Answer - b
Given equation can be written as
` x dy = ( sqrt(x^(2)+y^(2))+y)dx,`
I.e ` (dy)/(dx) = (sqrt(x^(2)+y^(2))+y)/x + y" "` …(i)
This is a homogenous differential equation . To simplify it ,
Put ` y - vx rArr (dy)/(dx) = v+x (dv)/(dx)`
` v+ x (dv)/(dx) = (sqrt(x^(2)+v^(2)x^(2))+vx)/x`
i.e `v + x (dv)/(dx) = sqrt(1+v^(2))+v`
` x (dv)/(dx) = sqrt(1+v^(2))`
` rArr (dv)/(sqrt(1+v^(2)))=(dx)/x" " ` ...(ii)
On integrating both sides of Eq. (ii) ,we get
` log ( v + sqrt(1+v^(2))) = log x + log C`
` rArr v + sqrt(1+v^(2))= Cx`
` rArr y/x + sqrt(1+(y^(2))/(x^(2)))= Cx`
`rArr y + sqrt(x^(2) + y^(2)) = Cx^(2)`