Solution of the differential equation `x(dy)/(dx)=y+sqrt(x^(2)+y^(2))`, is

Options

A. `y-sqrt(x^(2)+y^(2))=cx^(2)`
B. `y+sqrt(x^(2)+y^(2))=cx^(2)`
C. `y+sqrt(x^(2)+y^(2))=cy^(2)`
D. `x-sqrt(x^(2)+y^(2))=cy^(2)`

Answer

B

Explanation

Given `(dy)/(dx)=(sqrt(x^(2)+y^(2))+y)/x`
Put `y=vx`
`implies(dy)/(dx)=v+x (dv)/(dx)`
`:.v+x(dv)/(dx)=(sqrt(x^(2)+v^(2)x^(2))+vx)/x`
`implies(dv)/(sqrt(1+v^(2)))=(dx)/x`
`implies log(v+sqrt(1+v^(2)))=logx+logc`
[intregrating]
`implieslog(y/x+sqrt(1+(y^(2))/(x^(2))))=logcx`
`impliesy+sqrt(x^(2)+y^(2))=cx^(2)`