Final answer:
The initial-value problem involves solving a Bernoulli differential equation after simplification and using integration. With the given initial condition y(0) = 9, we determined the constant of integration and expressed the solution of the differential equation.
Step-by-step explanation:
To solve the initial-value problem with the Bernoulli differential equation y^(1/2) (dy/dx) y^(3/2) = 1, with the initial condition y(0) = 9, we start by simplifying the equation. We notice that y^(1/2) and y^(3/2) can be combined to y^2.
The simplified differential equation is dy/dx = 1/y^2. We can solve this equation by integrating both sides with respect to x. The integration of 1/y^2 with respect to y is -1/y, and the integration of 1 with respect to x is x. So, after integration, we will have -1/y = x + C, where C is the constant of integration.
Using the initial condition y(0) = 9, we can find the value of C. Substituting y = 9 and x = 0 into -1/y = x + C gives us C = -1/9. Now, with the value of C known, we can express the solution as y = -1/(x - 1/9).
The final step is to ensure that the function is in the correct domain, as we cannot have y equal to zero or less since it would not make sense in the context of the initial-value problem we are solving.