Solve the differential equation. \[ \frac{d y}{d x}=x \sqrt{y} \] for \( y \neq 0 \)
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To solve the differential equation \(\frac{d y}{d x} = x \sqrt{y}\), we can separate the variables. Rearranging gives us: \[ \frac{1}{\sqrt{y}} \, dy = x \, dx. \] Next, integrate both sides. The left side integrates to \(2\sqrt{y}\) and the right side integrates to \(\frac{1}{2}x^2\). So we have: \[ 2\sqrt{y} = \frac{1}{2} x^2 + C, \] where \(C\) is a constant of integration. Divide through by 2 to simplify: \[ \sqrt{y} = \frac{1}{4} x^2 + \frac{C}{2}. \] Now, squaring both sides gives us: \[ y = \left(\frac{1}{4} x^2 + \frac{C}{2}\right)^2. \] This represents the general solution to the differential equation.
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