Problem 35 - Advanced Engineering Mathematics

Question

Use the concept that \(y=c,-\infty

Step-by-Step Solution

Answer

The differential equation does not have constant solutions.

Step 1: Rewrite the differential equation

The problem gives us the equation \((y-1) y^{\prime}=1\). Here, \(y^{\prime}\) represents the derivative of \(y\) with respect to \(x\). Our goal is to examine if this equation has constant solutions, meaning solutions where the derivative is zero. Let's begin by rewriting the equation: \((y-1) y^{\prime}=1\).

Step 2: Analyze condition for constant solutions

A function is constant if its derivative is zero, that is, if \(y^{\prime}=0\). Substitute \(y^{\prime}=0\) into the equation: \((y-1) \cdot 0=1\). This simplifies to \(0=1\), which is a contradiction. Therefore, no constant solution satisfies this equation.

Step 3: Conclusion

Since substituting \(y^{\prime}=0\) yields a contradiction, the given differential equation \((y-1) y^{\prime}=1\) does not have any constant solutions. This result holds under the condition that a constant function must have a zero derivative.

Key Concepts

Understanding Differential Equations

A differential equation involves derivatives, which are expressions representing rates of change. Essentially, a differential equation tells us how a quantity changes in relation to another variable. For the function described in the original exercise, we have an expression

• \((y-1) y^{\prime}=1\).

Here, \(y\) is a function of another variable—often \(x\)—and \(y^{\prime}\) represents its derivative or rate of change with respect to \(x\).
Differential equations can model a wide array of phenomena:

• Physics: such as the motion of particles
• Biology: like population dynamics
• Economics: for forecasting trends

In this scenario, our task is to determine if there's a constant solution, which would mean \(y^{\prime}\) equals zero, ensuring no change over \(x\).

Exploring Constant Functions

Constant functions are simple yet important in mathematics. They are functions that do not change, regardless of the input value. Mathematically, a constant function is represented as \(y = c\), where \(c\) is a constant value.
When dealing with differential equations, a constant function implies that the derivative is zero. This is because:

• Derivatives signify change
• A zero derivative indicates no change

In this context, if \(y = c\), then \(y^{\prime}=0\). The exercise you encountered required checking whether a constant function is possible by substituting \(y^{\prime}=0\) into our equation \((y-1) y^{\prime}=1\). Since this leads to a contradiction or absurd result, it suggests that no constant solution—a function \(y\) remaining at a single value—exists for this differential equation.

Zero Derivative Insight

A zero derivative is a key concept in calculus, especially when analyzing whether a function remains unchanged across its domain. The condition \(y^{\prime}=0\) means that no matter how the independent variable \(x\) changes, the dependent variable \(y\) remains static.
A zero derivative can occur in various settings:

• Flat lines like horizontal segments on a graph
• Areas where change is momentarily paused

In exploring differential equations, this concept is crucial for constant solutions. Checking for a constant solution within our original differential equation, setting \(y^{\prime}=0\), leads quickly to an evident contradiction because it simplifies the equation \((y-1) \cdot 0=1\) to \(0=1\), an impossibility.
This tells us definitively that no part of \(y\) can remain unchanged (or constant) throughout its domain, reflecting how understanding zero derivatives can help identify such solutions.