Final answer:
To solve the differential equation y' = x²/y, we can begin by multiplying both sides by y and dx to separate variables. Next, integrate both sides with respect to their respective variables: ∫ydy = ∫x²dx. Finally, to solve for y, we can multiply both sides by 2 and take the square root: y = ±√((2/3)x³ + 2C).
Step-by-step explanation:
To solve the differential equation y' = x²/y, we can begin by multiplying both sides by y and dx to separate variables. This gives us ydy = x²dx. Next, we integrate both sides with respect to their respective variables: ∫ydy = ∫x²dx. Integrating ydy gives us (1/2)y², and integrating x²dx gives us (1/3)x³. Now we have (1/2)y² = (1/3)x³ + C, where C is the constant of integration. Finally, to solve for y, we can multiply both sides by 2 and take the square root: y = ±√((2/3)x³ + 2C).