Question

Experiments indicate that glucose is absorbed by the body at a rate proportional to the amount of glucose present in the bloodstream. Let \(\lambda\) denote the (positive) constant of proportionality. Now suppose glucose is injected into a patient's bloodstream at a constant rate of \(r\) units per unit of time. Let \(G=G(t)\) be the number of units in the patient's bloodstream at time \(t>0 .\) Then $$ G^{\prime}=-\lambda G+r $$ where the first term on the right is due to the absorption of the glucose by the patient's body and the second term is due to the injection. Determine \(G\) for \(t>0,\) given that \(G(0)=G_{0} .\) Also, find \(\lim _{t \rightarrow \infty} G(t)\)

Short answer

Answer: The number of units of glucose in the patient's bloodstream approaches $\dfrac{r}{\lambda}$ as time goes to infinity.

Step-by-step solution

Identify the ODE and write it in standard form

We have the following ordinary differential equation(ODE) given: $$G^{\prime}(t)=-\lambda G(t)+r$$ To solve this linear ODE, we can rewrite it in the standard form as: $$G^{\prime}(t) + \lambda G(t) = r$$ Now, let's find an integrating factor and multiply it to both sides of the equation.

Find the integrating factor

The integrating factor can be found by calculating the exponential of the integral of the coefficient of \(G(t)\), which in this case is \(\lambda\): Integrating Factor: \(\mu(t)=e^{\int \lambda dt} = e^{\lambda t}\) Multiply this integrating factor to both sides of the ODE: $$e^{\lambda t}(G^{\prime}(t)+\lambda G(t))=re^{\lambda t}$$

Simplify the ODE and integrate

Observe that the left side of the equation can be written as the derivative of the product of the integrating factor and \(G(t)\): $$\frac{d}{dt}(e^{\lambda t} G(t)) = re^{\lambda t}$$ Now, integrate both sides with respect to \(t\): $$\int \frac{d}{dt}(e^{\lambda t} G(t)) dt = \int re^{\lambda t} dt$$ This simplifies to: $$e^{\lambda t}G(t) = \frac{r}{\lambda}e^{\lambda t} + C$$

Solve for G(t) using initial conditions

Divide both sides of the equation by \(e^{\lambda t}\) and use the initial condition \(G(0)=G_{0}\) to find the constant \(C\): $$G(t)=\frac{r}{\lambda} + Ce^{- \lambda t}$$ $$G_0=\frac{r}{\lambda} +C\cdot e^0 \implies C=G_0-\frac{r}{\lambda}$$ Plugging the value of C back in, we get the expression for \(G(t)\): $$G(t) = \frac{r}{\lambda} +(G_0 -\frac{r}{\lambda})e^{- \lambda t}$$

Calculate the limit as t approaches infinity

To find \(\lim _{t \rightarrow \infty} G(t)\), we can take the limit of the expression we found for \(G(t)\): $$\lim _{t \rightarrow \infty} \left(\frac{r}{\lambda} +(G_0 -\frac{r}{\lambda})e^{- \lambda t}\right)$$ As t goes to infinity, the exponential term \(e^{-\lambda t}\) converges to zero. Therefore, the limit is: $$\lim _{t \rightarrow \infty} G(t) = \frac{r}{\lambda}$$ So, the number of units of glucose in the patient's bloodstream approaches \(\dfrac{r}{\lambda}\) as time goes to infinity.

Key concepts

Linear ODE

Linear ordinary differential equations (ODEs) are a type of differential equation characterized by linearity in the unknown function and its derivatives. In essence, linear ODEs can be represented in the form \[\begin{equation}a_n(t) \frac{d^n y}{dt^n} + a_{n-1}(t) \frac{d^{n-1} y}{dt^{n-1}} + \text{...} + a_1(t) \frac{dy}{dt} + a_0(t) y = g(t),\text{\end{equation}\]} where each \( a_i(t) \) is a function of \( t \) (possibly constant), \( y \) is the unknown function to be solved for, and \( g(t) \) is a given function. When \( g(t) = 0 \), the equation is called homogeneous; otherwise, it is nonhomogeneous, as is the case in the presented exercise where the nonhomogeneous term is \( r \), a constant rate of glucose injection.

Integrating Factor Method

The Integrating Factor Method is a technique commonly used to solve first-order linear ODEs. This method can be particularly helpful when the differential equation is nonhomogeneous. The magic of the integrating factor lies in its ability to turn the left-hand side of the ODE into the derivative of a product, making it simpler to integrate. The integrating factor, \( \boldsymbol{\mu(t)} \), is usually calculated as \( e^{\text{\int} P(t)\text{dt}} \), where \( P(t) \) is the coefficient of \( y \) in the ODE rewritten in the form \( y' + P(t)y = Q(t) \).By multiplying throughout by the integrating factor, the ODE becomes an exact differential, and in our exercise, it transformed the complicating variable coefficients into a perfect derivative, which was then straightforward to integrate.

Exponential Growth and Decay

Exponential growth and decay are mathematical concepts that reflect processes that increase or decrease at rates proportional to the current value. Many real-world phenomena, such as population dynamics, radioactive decay, and, as in our exercise, the absorption rate of a drug in the bloodstream, can be modeled using equations that involve an exponential function.The glucose absorption rate, characterized by the term \( -\boldsymbol{\backslash lambda G} \), demonstrates exponential decay, with \( \boldsymbol{\backslash lambda} \) being the constant of proportionality. As time progresses, the quantity \( G(t) \) will decrease ever more slowly and never completely reach zero but rather approach an asymptotic value, owing to the continuous nature of the exponential decay process.

Limits in Calculus

In calculus, the concept of limits helps us understand the behavior of functions as they approach a certain point. They are foundational for defining derivatives and integrals. A limit examines the value that a function \( f(x) \) approaches as \( x \) comes close to a certain point. In our example, we computed the limit as \( t \) approaches infinity for \( G(t) \) to determine the long-term behavior of the glucose in the patient's bloodstream.The computation showed that the exponential term \( (G_0 -\frac{r}{\boldsymbol{\backslash lambda}})e^{- \boldsymbol{\backslash lambda t}} \) in our solution diminishes over time, implying that the glucose level would stabilize at \( \frac{r}{\boldsymbol{\backslash lambda}} \), demonstrating how limits provide insight into system behavior that might not be immediately obvious. They are essential for understanding long-term tendencies in dynamic systems.