Question

on your own, verify that the indicated expression is an implicit solution of the given first-order differential equation. dx dt = (x − 1)(1 − 2x); ln 2x − 1 x − 1 = t

Answer 1

The student is verifying that a given logarithmic expression is a solution to a first-order differential equation by differentiating the expression with respect to time, using the chain rule, separating variables, and integrating to show that it produces the original differential equation.

Step-by-step explanation:

The student is tasked with verifying that the given expression ln (2x − 1)/(x − 1) = t is an implicit solution to the differential equation dx/dt = (x − 1)(1 − 2x). This involves showing that by differentiating the given expression with respect to time t, one arrives back at the original differential equation. The confirmation uses techniques such as the chain rule for differentiation, separation of variables, and integration.

To verify the solution, one would start by differentiating the expression with respect to t, which indirectly involves differentiating with respect to x since x is a function of t. By performing this differentiation correctly and simplifying, we should arrive back at the given differential equation, thus confirming that the initial expression is indeed an implicit solution.

Throughout this process, one would apply the following steps:

1. Differentiate the given expression with respect to t using the chain rule.
2. Separate the variables x and t as needed.
3. Integrate as required to achieve the original differential equation.