please solve this differential equation by "Bernoulli's Equation" y+x(dy/dx)=(x)(square root of y). thank you in advance.     

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to write the Bernoulli standard form of differential equation such that:

`y' + P(x)y = Q(x)y^n`

Comparing the standard equation to the given form `y + x(dy)/(dx) = xsqrty` , you need to divide by `x sqrt y`  both sides such that:

`y/(x sqrt y) + (dy)/(dx)/sqrty = 1`

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

You should come up with the substitution `p = sqrt y =gt dp = 1/(2 sqrt y)*(dy)/(dx)`

You need to substitute `p`  for `sqrt y`  and dp for `1/(2 sqrt y)*(dy)/(dx) ` such that:

`p/x + 2dp = 1 `

Dividing by 2 both sides yields:

`p/(2x) + dp = 1/2`

You need to find integrating factor `mu(x) = e^int (dx)/(2x) = e^((1/2)*ln x)` `mu` (x)=`e^(ln sqrt x) =gt mu (x) = sqrt x`

`int sqrt x*dp = int (sqrt x dx)/2`

`sqrt x*p = (1/3)*x sqrt x + c`

`p(x) = (1/3)x + c/sqrt x`

You need to substitute `sqrt y`  for `p`  such that:

`sqrt y = (1/3)x + c/sqrt x`

`y = (x^2)/3 + c/x`

Hence, evaluating the general solution to differential equation, using Bernoulli's equation, yields `y = (x^2)/3 + c/x.`