Solve the following differential equations :
(1) ` (dy)/(dx) = (y + sqrtx^(2)+y^(2))/x `

Explanation

`(dy)/(dx)=(y+sqrt(x^(2)y^(2)))/(x)" …(i)"`
Putting `y=vx rArr (dy)/(dx)=v+x(dv)/(dx)`
`therefore" "v=(y)/(x)`
`therefore" "` Equation (i) becomes,
`v+x(dv)/(dx)=(vx+sqrt(x^(2)+v^(2)x^(2))/(x))`
`therefore" "v+x(dv)/(dx)=v+sqrt(1+v^(2))`
`rArr" "x(dv)/(dx)=sqrt(1+v^(2))`
`rArr" "(1)/(sqrt(1+v^(2)))dv=(1)/(x)dx`

On integration, we get
`int(1)/(sqrt(1+v^(2)))dv=int(1)/(x)dx`
`rArr" "log lv+sqrt(1+v^(2))|=log|x|+logx`
`rArr" "log|(y)/(x)+sqrt(1+(y^(2))/(x^(2)))|=log|cx|`
`rArr " "y+sqrt(x^(2)+y^(2))=cx^(2)`
This is the general solution.