Final answer:
To solve the separable differential equation dx/dt = x² + 1/64, separate the variables and integrate. Use the initial condition x(0) = 9 and find the particular solution x(t) = -1/8 ± e^(t + ln(72) - ln(|2e^3 - 1|)).
Step-by-step explanation:
To solve the separable differential equation dx/dt = x² + 1/64, we can separate the variables and integrate. Rearranging the equation, we get dx/(x² + 1/64) = dt. Integrating both sides, we have ln|x + 1/8| = t + C. Solving for x, we get x = -1/8 ± e^(t + C). Using the initial condition x(0) = 9, we can substitute the values into the equation to find the particular solution.
Substituting x = 9 and t = 0 into the equation, we have -1/8 ± e^C = 9. Solving for C, we find C = ln(72) - ln(|2e^3 - 1|). Hence, the particular solution satisfying the initial condition is x(t) = -1/8 ± e^(t + ln(72) - ln(|2e^3 - 1|)).