Question

Write and solve the differential equation that models the verbal statement. The rate of change of \(Q\) with respect to \(t\) is inversely proportional to the square of \(t\).

Short answer

The differential equation that models the given verbal statement is: dQ/dt = k/t². The solution to this differential equation is: Q = -k/t + C, where k and C are constants.

Step-by-step solution

Write the Differential Equation

From the statement, 'The rate of change of Q with respect to t is inversely proportional to the square of t', we can derive the mathematical equivalent. The rate change of Q with respect to t can be represented as dQ/dt. 'Inversely proportional to the square of t' can be represented as 1/t². Therefore, the differential equation that models the verbal statement is given as: dQ/dt = k/t², where k is a constant.

Solve the Differential Equation

This is a separable differential equation which can be rearranged and solved by integrating both sides. The equation can be rewritten as dQ = k (1/t²) dt. Then, integrate both sides of the equation. For the left side, the integral of dQ is just Q. For the right side, integrating k (1/t²) dt gives -k/t since the integral of t^-2 is -t^-1. Therefore, upon integration, the solution to the differential equation is: Q = -k/t + C, where C is the constant of integration.