Solve the given differential equation by separation of variables. `y(4-x^2)^(1/2) dy = (4+y^2)^(1/2) dx`

Solve the given differential equation by separation of variables.

`y(4-x^2)^(1/2) dy = (4+y^2)^(1/2) dx`

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Solution:

To solve the differential equation `y(4-x^2)^(1/2) dy = (4+y^2)^(1/2) dx` 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 `(4+y^2)^(1/2)`

`y(4-x^2)^(1/2) dy/(4+y^2)^(1/2) = dx`

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:

`∫y(4-x^2)^(1/2) dy = ∫dx/(4+y^2)^(1/2) + C`

Integrating both sides with respect to their respective variables gives:

`∫y(4-x^2)^(1/2) dy = x + C`

Solving for y gives:

`y(4-x^2)^(3/2) = x + C`

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So, the general solution is `y(4-x^2)^(3/2) = x + C` where C is an arbitrary constant.