The step-by-step solution is as follows:
1. The problem starts with a first-order ordinary differential equation (ODE): dx/dt = x/9. This equation is separable, meaning we can separate the variables x and t on opposite sides of the equation. In this form, the equation can be easily solved by integration.
2. We keep dx on one side and move dt to the other side leading to: dx/x = dt/9. This is possible because of the homogeneous property of the equation where we can divide by x(t) on both sides.
3. Now, we proceed to integrate both sides of the equation. The integration of dx/x is ln|x|. On the other side, the integration of dt/9 is t/9. So, after integration, we have: ln|x| = t/9 + C, where C is the constant of integration.
4. To remove the natural logarithm, we take the exponent of both sides which gives: |x| = e^(t/9+C).
5. For simplicity, we write e^C as another constant C1. So, the general solution becomes: x(t) = C1*e^(t/9). This tells us what the function x(t) looks like for all time t.
6. To find the particular solution, we use the given initial condition, x(0) = 8. We substitute these values into the equation: 8 = C1*e^(0). As any number to the power of 0 equals 1, this yields us C1 = 8 .
7. Therefore, the particular solution of our differential equation that fulfills the initial condition x(0) = 8 is: x(t) = 8*e^(t/9).
Remember that this is an exponential function, which means x(t) grows ever larger as t increases, but also that it does so slowly, as the growth factor is quite small (1/9).