Question

The solution of `sqrt(x)*y' + (1-2sqrt(x))*sqrt(y)= 0` is `(-2*sqrt(x)-1)*e^-sqrt(x)` ??

Answer 1

No. Please see the explanation.

Explanation:

Given: `sqrt(x)*y' + (1-2sqrt(x))sqrt(y)= 0`

Use the separation of variables method:

`sqrt(x)dy/dx = (2sqrt(x)-1)sqrt(y)`

`1/sqrtydy = (2-1/sqrtx)dx`

`2sqrty= 2x-2sqrtx+C`

`sqrty= x-sqrtx+C`

`y = (x-sqrtx+C)^2 larr` this is not the same form as the proposed solution, therefore, the answer is no.

Answer 2

`y = (x-sqrtx + C_0)^2`

Explanation:

Making

`z = sqrty rArr z' = 1/2 (y')/sqrty rArr y' = 2 z z'`

so

`sqrt(x)*y' + (1-2sqrt(x))*sqrt(y)= 0 equiv 2sqrt(x)*zz' + (1-2sqrt(x))*z= 0 rArr z(2sqrt(x)z' + (1-2sqrt(x)))= 0`

and then

`{(z=0),(2sqrt(x)z' + (1-2sqrt(x))= 0):}`

or

`z' = -(1-2 sqrtx)/(2sqrtx) rArr z = x-sqrtx + C_0`

and finally

`y = (x-sqrtx + C_0)^2`