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(Solved): Solve the equation f(x)=0 to find the critical points of the given autonamous differential equation ...
Solve the equation to find the critical points of the given autonamous differential equation . Analyze the sign of f(x) to delermine whether each critical point is stable or unstablo, and construct the corresponding phase diagram for the differential equation. Solve the differential equation explicitly for x(t) in terms of t. Finally, use either the exact solution or a computer-generated slope field to sketch typical solution curves for the given differential equation, and verify visually the stability of each critical point:
Expert Answer
To find the critical points of the autonomous differential equation, we need to solve the equation where Setting we have:[x - 1 = 0]Solving for (x):[x = 1]So, the critical point of the differential equation is (x = 1).To analyze the stability of the critical point, we examine the sign of (f(x)) around the critical point.For which means the function is negative. This indicates that the solutions will move towards the critical point (x = 1) as (t) increases, suggesting stability.For (x > 1), (f(x) = x - 1 > 0), which means the function is positive. This indicates that the solutions will move away from the critical point (x = 1) as (t) increases, suggesting instability.