Solve the given differential equation by separation of variables. `(x+ \sqrt(x)) dy/dx = y + \sqrt(y) `

Solve the given differential equation by separation of variables.  

`(x+ \sqrt(x)) dy/dx = y + \sqrt(y) `

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Solution:

To solve the differential equation `(x+ \sqrt(x)) dy/dx = y + \sqrt(y)` by separation of variables, we first need to rewrite it in the form `f(x)dx = g(y)dy`. To do this, we can divide both sides by y

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

Now we can separate the variables by moving all the y terms to one side and all the x terms to the other side. This gives us:

`∫(x+ \sqrt(x)) dy = ∫ (1 + \sqrt(y)) dx + C`

Integrating both sides with respect to their respective variables gives:

`x*y + \int\sqrt(x) dy = x + 2*\int\sqrt(y) dy + C`

Solving for y gives:

`y(x+\sqrt(x))=x+2y^(3/2)+C`

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So, the general solution is `y(x+\sqrt(x))=x+2y^(3/2)+C` where C is an arbitrary constant.