Final Answer:
The solution to the initial value problem is y(x) = (x^2 + 81)^(1/2).
Step-by-step explanation:
The differential equation given is in a particular form that can be solved by separating variables. To solve this equation, first, rewrite it in a form suitable for separation. Rearrange the terms to have y and dy on one side and x and dx on the other side. The equation can then be manipulated into a form that allows for integrating both sides separately.
After rearranging the terms and performing the integration, the equation is transformed to y^(3/2)/3 - y^(1/2) = x + C, where C is the constant of integration. Applying the initial condition y(0) = 9, we can solve for the constant, C.
Plugging in the initial condition y(0) = 9, we can solve for C, which turns out to be 0. Substituting C back into the equation obtained from the integration, we get y^(3/2)/3 - y^(1/2) = x. Solving this equation for y yields the final solution y(x) = (x^2 + 81)^(1/2).
Therefore, the solution to the initial value problem is y(x) = (x^2 + 81)^(1/2). This solution satisfies the given differential equation and the initial condition y(0) = 9.