Final answer:
To find an explicit solution for the initial-value problem dx/dt = 8(x² - 1), x(4) = 1, we can separate variables, integrate, and solve for x(t). The explicit solution is x(t) = (x+1)/(x-1) = Ae^(16t), where A is a constant.
Step-by-step explanation:
To find an explicit solution for the given initial-value problem dx/dt = 8(x² - 1), x(4) = 1, we can separate variables and integrate both sides of the equation.
Starting with the differential equation:
dx/dt = 8(x² - 1)
We can rewrite it as:
dx / (x² - 1) = 8 dt
Integrating both sides gives:
∫dx / (x² - 1) = ∫8 dt
Now, we can evaluate the integrals:
1/2 ln |x+1| - 1/2 ln |x-1| = 8t + C
Simplifying and solving for x(t):
ln |(x+1)/(x-1)| = 16t + C
Now, we can raise both sides as a power of e:
|(x+1)/(x-1)| = e^(16t + C)
Take note that the absolute value can be neglected because it doesn't affect the solution. So we can rewrite it as:
x(t) = (x+1)/(x-1) = Ae^(16t)
Where A = ±e^C.