From: drhuang57@gmail.com   
      
   On Saturday, 9 July 2022 at 17:04:16 UTC+10, Nasser M. Abbasi wrote:   
   > On 7/9/2022 1:43 AM, Dr Huang wrote:   
   > > wolfram gives   
   > > (c1-x)^2/x   
   > > is it right? did it seem wrong? what is yours?   
   > >   
   > > DrHuang.com   
   > Mathematica 13.1   
   >   
   > ClearAll[x, y]   
   > ode = x*y'[x] + y[x] == 2*(x*y[x])^(1/2);   
   > sol = DSolve[ode, y[x], x, IncludeSingularSolutions -> True]   
   >   
   > gives   
   >   
   > {{y[x] -> (E^(C[1]/2) + x)^2/x}, {y[x] -> x}}   
   >   
   > The first is the general solution and the second is singular solution.   
   >   
   > The solution you give is a particular solution which satisfies the ode   
   > under the condition x>=C[1] only. You can see this by doing   
   >   
   > sol = y -> Function[{x}, (C[1] - x)^2/x]   
   > result = ode /. sol // FullSimplify   
   >   
   > Sqrt[(x - C[1])^2] + C[1] == x   
   >   
   > The above becomes identity when x >= C[1]   
   >   
   > ps. I do not use Wolfram Alpha.   
   >   
   > --Nasser   
      
   thanks   
   DrHuang.com   
      
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