Question

Find an explicit solution of the given initial-value problem. dx/dt = 8(x² - 1), x(4) = 1?

Answer 1

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.