Solve the given differential equation subject to the indicated initial condition. `ydy = 4x(y^2 + 1 )^(1/2) dx , y(0) = 1`

Solve the given differential equation subject to the indicated initial condition. 

`ydy = 4x(y^2 + 1 )^(1/2) dx , y(0) = 1`

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Solution:

To solve the differential equation `y dy = 4x(y^2 + 1 )^(1/2) dx` with the initial condition `y(0) = 1`, we can first use the separation of variables to get:

`∫ y dy = ∫ 4x(y^2 + 1 )^(1/2) dx`

Integrating both sides with respect to their respective variables gives:

`(1/2)y^2 = 2x(y^2 + 1 )^(3/2) + C`

Solving for y gives:

`y^2 = 4x(y^2 + 1 )^(3/2) + 2C`

Now we can use the initial condition to find the value of C:

`y(0) = 1 = 4*0*(1 + 1)^(3/2) + 2C = 2C`

`C = 1/2`

So the general solution is:

`y^2 = 4x(y^2 + 1 )^(3/2) + 1`

And the solution to the differential equation subject to the initial condition is:

`y^2 = 4x(y^2 + 1 )^(3/2) + 1`

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It's important to note that this differential equation does not have a closed-form solution in terms of elementary functions. The result is an implicit solution, which can be solved graphically or numerically.