Final answer:
The general solution to the differential equation x'(t) = 2tx² is found by separating variables and integrating, resulting in -1/x = t² + C. With initial condition x(0) = 1, the particular solution is -1/x = t² - 1, and the maximum interval of existence is (-1, 1).
Step-by-step explanation:
To find the general solution of the differential equation x'(t) = 2tx², we need to separate the variables and integrate. This is also known as solving a separable differential equation. By dividing both sides of the equation by x² and then multiplying by dt, we have dx/x² = 2tdt. After integrating both sides, we get -1/x = t² + C, where C is the constant of integration. To find C, we can use the initial condition x(0) = 1.
Plugging in the initial values, we find that C = -1. This means the particular solution satisfying the initial condition is -1/x = t² - 1. The maximum interval of existence for this solution is determined by the point where the solution becomes undefined, which is when x(t) → 0. So, we solve t² - 1 = 0, and we find t = ±1. Therefore, the maximum interval of existence is (-1, 1).