Final answer:
The solution of the differential equation y′=x²y is y = e^[(1/3)x³ + C - C1].
Step-by-step explanation:
The given differential equation is y' = x²y.
To solve this equation, we can separate the variables and integrate both sides.
1. Move all the terms involving y to the left-hand side and all the terms involving x to the right-hand side: 1/y dy = x² dx.
2. Integrate both sides: ∫(1/y) dy = ∫x² dx.
3. The integral of 1/y with respect to y is ln|y| + C1, where C1 is the constant of integration. The integral of x² with respect to x is (1/3)x³ + C2, where C2 is the constant of integration.
4. Equate the expressions involving the constants of integration: ln|y| + C1 = (1/3)x³ + C2.
5. Simplify the equation and solve for y: ln|y| = (1/3)x³ + C - C1, where C = C2 - C1. Taking the exponential of both sides, we get |y| = e^[(1/3)x³ + C - C1].
6. Since |y| is always positive, we can remove the absolute value sign: y = e^[(1/3)x³ + C - C1].
Therefore, the solution to the given differential equation is y = e^[(1/3)x³ + C - C1].