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

Explanation

` (dy)/(dx) = y + sqrt(x^(2) +y^(2))/x `
Put y =vx ` therefore (dy)/(dx) =v +x (dv)/(dx)`
(1) becomes ` v + x(dv)/(dx) = vx +sqrt(x^(2)+v^(2)x^(2))/x`
` v+ x (dv)/(dx) = v + sqrt(1 +v^(2))`
` x(dv)/(dx) =sqrt(1+v^(2)) " " therefore 1/(sqrt(1+v^(2)) dv = 1/x dx`
Integrating we get,
` int 1/(sqrt(1 +v^(2)) dv = int 1/x dx +c_(1)`
` therefore log|v+ sqrt(1+v^(2)) | = log |x| + log c, " where " c_(1) = log c`
` log |y/x + sqrt(1+y^(2)/x^(2)| = log |cx|`
` (y + sqrt(x^(2)+y^(2)))/x =cx " " therefore y + sqrt(x^(2) +Y^(2)) =cx^(2) `
This is the general solution.