{Use of Tech} Intravenous drug dosing The amount of drug in the blood of a patient (in milligrams) administered via an intravenous line is governed by the initial value problem y’(t) = -0.02y + 3, y(0) = 0 where t is measured in hours.


b. What is the steady-state level of the drug?

Verified step by step guidance

1. Recognize that the steady-state level of the drug corresponds to the equilibrium solution where the amount of drug no longer changes over time. Mathematically, this means setting the derivative \(y'(t)\) to zero: \$0 = -0.02y + 3$.
2. Solve the equation \$0 = -0.02y + 3\( for \)y\( to find the steady-state concentration. Rearrange it to isolate \)y\(: \)0.02y = 3$.
3. Divide both sides of the equation by \$0.02\( to solve for \)y\(: \)y = \frac{3}{0.02}$.
4. Interpret this value as the long-term amount of drug in the blood when the system reaches equilibrium, meaning the drug input and elimination balance out.
5. Note that this steady-state value is independent of the initial condition \(y(0) = 0\) because it represents the behavior as \(t \to \infty\).

Key Concepts

First-Order Linear Differential Equations

This type of differential equation has the form y' + p(t)y = q(t). It models processes where the rate of change depends linearly on the current state and an external input. Solving such equations involves finding an integrating factor or using steady-state analysis.

Steady-State Solution

The steady-state level is the long-term behavior of the system when the rate of change becomes zero (y' = 0). It represents the equilibrium concentration where the drug input balances the elimination, providing a constant drug level in the blood.

Initial Value Problem (IVP)

An IVP specifies a differential equation along with an initial condition, y(0) = y0, which allows for a unique solution. It models real-world scenarios where the starting state is known, such as the initial drug concentration at time zero.