Final answer:
To solve the given initial-value problem, we can use the substitution method. Divide both sides of the equation by y^(3/2) and substitute u = y^(1/2) to transform the equation into a linear differential equation. Solve the resulting differential equation to find the solution for u, and then substitute back u = y^(1/2) to find the solution for y.
Step-by-step explanation:
To solve the given initial-value problem, we can use the substitution method. First, let's divide both sides of the equation by y^(3/2) to get y^(1/2) * dy/dx - 1/y^(1/2) = 1/y^(3/2).
Let's substitute u = y^(1/2), which means du/dx = (1/2) * y^(-1/2) * dy/dx. Plugging these substitutions into the equation, we get (1/2) * du/dx - 1/u = 1/u^3.
Now, the equation is in the form of a linear differential equation. Let's solve it by multiplying both sides by 2u^3 to eliminate the fraction. This gives us u^4 * du/dx - 2u^2 = 2.
We can solve this differential equation using standard techniques like separation of variables or integrating factors. After finding the solution for u, we can substitute back u = y^(1/2) to find the solution for y.