Problem 78
Question
The differential equation \(y^{\prime}=-a y \ln (y / b)\) (for positive constants \(a\) and \(b\) ) arises in the study of the growth of some animal tumors. Solve the differential equation and sketch several members of the family of solutions. What adjective (e.g., rapid, moderate, slow) would you use to characterize this type of growth?
Step-by-Step Solution
Verified Answer
The family of solutions for the given differential equation is \(y = be^{e^{-ax + c}}\) and this corresponds to a 'slow' growth.
1Step 1: Separate the Variables
Rearrange the given differential equation \(y' = -ay \ln(y/b)\) to separate the variables. This will result in \(\frac{dy}{y \ln(y/b)} = -a dx \).
2Step 2: Integrate Both Sides
Integrate both sides to find the solution, so the integral form is \(\int \frac{1}{y \ln(y/b)} dy = -a \int dx\). After integration, we obtain \(\ln|\ln(y/b)| = -ax + c\), where c is the constant of integration.
3Step 3: Solve for y
Take the exponential of both sides to isolate y. The result is \(\ln(y/b) = e^{-ax + c}\), and solving for y gives \(y = be^{e^{-ax + c}}\). This represents the family of solutions for the given differential equation.
4Step 4: Sketch Several Members of the Family of Solutions
While specific sketching instructions are difficult to provide without a specific context or mechanism, plotting several instances of \(y = be^{e^{-ax + c}}\) with different values of c on the same axis will give you a family of solution curves. In general, the curve tends to simplify as y gets larger, eventually resembling a relatively straight line.
5Step 5: Characterize the Growth
By observing the sketches, it can be concluded that the growth rate of the function decreases as x increases, thus we can characterize the growth as 'slow'.
Key Concepts
Separation of VariablesExponential GrowthIntegration of Functions
Separation of Variables
When faced with a differential equation like \(y^{\textprime} = -ay \ln(y/b)\), the method of separation of variables allows us to find a solution by manipulating the equation so that each variable appears on opposite sides. To achieve this, we typically rearrange the terms to isolate \(dy\) and \(dx\) on different sides of the equation. This is a fundamental step in solving many types of first-order differential equations.
For our example, we start by dividing both sides by \(y \ln(y/b)\) to get \(\frac{dy}{y \ln(y/b)} = -a dx\). This gives us a clear path to integrate each side separately, which leads to the solution of the differential equation. The beauty of separation of variables lies in its simplicity and mechanical nature, allowing us to solve a potentially complex problem systematically.
For our example, we start by dividing both sides by \(y \ln(y/b)\) to get \(\frac{dy}{y \ln(y/b)} = -a dx\). This gives us a clear path to integrate each side separately, which leads to the solution of the differential equation. The beauty of separation of variables lies in its simplicity and mechanical nature, allowing us to solve a potentially complex problem systematically.
Exponential Growth
Exponential growth is a process where the rate of increase of a quantity is proportional to the amount present. It is characterized by the presence of a quantity increasing at a rate that becomes rapidly more significant according to a constant ratio. In biology, technology, and finance, exponential growth often models how populations, data, and investments increase over time.
In the context of our differential equation, the solution \(y = be^{e^{-ax + c}}\) hints at a modified exponential behavior. Initially, as \(x\) is small, we observe a rapid increase in \(y\), which is consistent with exponential growth. However, due to the presence of the negative exponent and the logarithmic function, the growth rate diminishes as \(x\) increases. This results in what is known as 'slow' growth and is different from classic exponential growth, where the growth rate would continue to increase indefinitely. Thus, understanding the nature of exponential functions is crucial in characterizing the solution's behavior over time.
In the context of our differential equation, the solution \(y = be^{e^{-ax + c}}\) hints at a modified exponential behavior. Initially, as \(x\) is small, we observe a rapid increase in \(y\), which is consistent with exponential growth. However, due to the presence of the negative exponent and the logarithmic function, the growth rate diminishes as \(x\) increases. This results in what is known as 'slow' growth and is different from classic exponential growth, where the growth rate would continue to increase indefinitely. Thus, understanding the nature of exponential functions is crucial in characterizing the solution's behavior over time.
Integration of Functions
Integration is a core operation in calculus, often seen as the reverse process of differentiation. It is used to find the area under a curve, determine the accumulated quantity over time, or solve differential equations, among other applications. When we integrate a function, we are essentially summing an infinite number of infinitesimally small quantities.
In the given problem, we integrate the left-hand side \(\int \frac{1}{y \ln(y/b)} dy\) and the right-hand side \(-a \int dx\). Performing these integrals is usually not straightforward and may require substitution or other techniques to find a solution. The resulting expressions from the integration process give us valuable insights into the behavior of the function and allow us to work towards the general solution of the differential equation. It's worth noting that integrals don't always produce simple formulas; sometimes, they result in more complex expressions or involve special functions, indicative of the original equation's complexity.
In the given problem, we integrate the left-hand side \(\int \frac{1}{y \ln(y/b)} dy\) and the right-hand side \(-a \int dx\). Performing these integrals is usually not straightforward and may require substitution or other techniques to find a solution. The resulting expressions from the integration process give us valuable insights into the behavior of the function and allow us to work towards the general solution of the differential equation. It's worth noting that integrals don't always produce simple formulas; sometimes, they result in more complex expressions or involve special functions, indicative of the original equation's complexity.
Other exercises in this chapter
Problem 75
The downward velocity of a falling object is modeled by the differential equation \(\frac{d v}{d t}=32-0.4 v^{2} .\) If \(v(0)=0 \mathrm{ft} / \mathrm{s},\) the
View solution Problem 76
Suppose that \(f\) is a function such that \(f(x) \geq 0\) and \(f^{\prime}(x)0 .\) Show that the area of the triangle with sides \(x=0, y=0\) and the tangent l
View solution Problem 73
For the logistic equation \(y^{\prime}(t)=k y(M-y),\) show that a graph of \(\frac{1}{y} y^{\prime}\) as a function of \(y\) produces a linear graph. Given the
View solution