Final answer:
The given equation is a separable differential equation, which can be solved by separating the variables and integrating both sides.
Step-by-step explanation:
The given equation is a separable differential equation, which can be solved by separating the variables and integrating both sides.
Step-by-step solution:
1. Rearrange the equation to separate the variables: dy/(y(y² - 4)) = dx/2.
2. Partial fraction decomposition can be used to integrate the left-hand side: (A/(y-2) + B/(y+2))dy = dx/2.
3. Integrate both sides:
(A ln|y-2| + B ln|y+2|)/2 = x + C
4. Apply the initial condition y(0) = 3 to solve for the constants A and B.
5. Substitute the values of A and B into the equation and solve for y in terms of x.