Final answer:
The general solution to the given differential equation is w = C(x¹²+1). We can find this solution by separating variables and integrating both sides of the equation. After solving a quadratic equation, we can substitute back to find the general solution for w.
Step-by-step explanation:
The general solution to the given differential equation x² dw/dx = √(w) x² is: w = C(x¹²+1).
We can solve this by separating variables and integrating both sides of the equation. Firstly, we can move the x² term to the denominator of the right side to get: dw/√(w) = dx/x². Then we can integrate both sides: ∫dw/√(w) = ∫dx/x². The integral of dw/√(w) is 2√(w), and the integral of dx/x² is -1/x. Therefore, we get: 2√(w) = -1/x + C', where C' is the constant of integration.
Next, we can isolate w by squaring both sides of the equation: 4w = 1/x² - 2C'√(w). Rearranging the terms, we get: 4w + 2C'√(w) = 1/x². This is a quadratic equation in terms of √(w), so we can substitute √(w) with a new variable t. By substituting, we get: 4t² + 2C't - 1/x² = 0. Now we can solve this quadratic equation to find the value of t, and then substitute back to find w. After solving, we get: w = C(x¹²+1) as the general solution.