Problem 6 - Calculus: Early Transcendentals

Question

Identify the constant solutions (if any) of \(y^{\prime}=t e^{y}\).

Step-by-Step Solution

Answer

Constant solutions exist for any constant value of \\(y\\) at \\(t = 0\\).

Step 1: Understanding Constant Solutions

A constant solution is a solution where the derivative part, \(y'\), is equal to zero. This means the function does not change with respect to \(t\), and \('y'\) will be a constant.

Step 2: Setting the Equation for Constant Solutions

Since a constant solution implies \(y' = 0\), we can substitute this into the differential equation \(y' = t e^{y}\), leading to the equation \(0 = t e^{y}\).

Step 3: Solving the Constant Solutions Equation

For the product of \(t e^{y}\) to be zero, either \(t = 0\) or \(e^{y} = 0\) must be true. Since \(e^{y} > 0\) for all real values of \(y\), the only possible solution is when \(t = 0\).

Step 4: Identifying the Constant Solution

Given that \(t = 0\), \(y\) can be any constant value because it doesn't affect the equation when multiplied by zero. Thus, all functions \(y = c\), where \(c\) is any constant, are solutions when \(t = 0\).

Key Concepts

Constant Solutions

A constant solution in the context of differential equations occurs when the derivative of the function, denoted as \(y'\), equals zero. In simple terms, this means that the value of \(y\) doesn't change over time or with respect to the variable \(t\). It remains fixed, hence the term "constant." For example, consider a situation like the equation \(y' = te^{y}\). For \(y\) to be constant, the derivative must equal zero, which means the whole expression cannot change regardless of the value of \(t\) or \(y\).

To find a constant solution, you set the entire differential equation to zero. This method roots from the idea that the derivative, representing change, halts when zeroed out, leaving the function constant. The analysis of constant solutions simplifies many complex equations by narrowing down the possible nature of solutions.

For instance, substituting \(y' = 0\) into the equation \(y' = te^{y}\) results in \(0 = te^{y}\). The conclusion is that \(t\) must be zero because the expression \(e^{y}\) is never zero for real values of \(y\). Hence, every constant \(y = c\) is a solution when \(t = 0\).

Derivative

The concept of a derivative is central in calculus, serving as a tool to describe how a function changes at any given point. It is often written as \(y'\) or \(\frac{dy}{dt}\). Understanding derivatives involves grasping how they provide a function's rate of change or slope at a point. They're particularly important in differential equations, where they appear to represent the dynamics of systems.

For the equation \(y' = te^{y}\), the derivative \(y'\) describes how \(y\) changes with respect to \(t\). If \(y' = 0\), as in finding constant solutions, it indicates that \(y\) does not change as \(t\) changes, implying \(y\) is a constant function. This situation leads us to explore constant solutions for the differential equation.

Calculating derivatives is a process of applying differentiation rules, which may involve basic rules (like the power rule) or more complex methods (like the chain rule), depending on the function. In differential equations, understanding these changes helps in modeling behaviors and solving real-world problems.

Exponential Function

The exponential function, expressed as \(e^{y}\), is a mathematical function noted for its constant rate of growth or decay. It is a fundamental function in mathematics due to its unique property of being its own derivative. The constant \(e\) is approximately 2.71828 and arises naturally in many areas of mathematics, including compound interest, probability theory, and complex number theory.

In the context of the differential equation \(y' = te^{y}\), the exponential function \(e^{y}\) influences how \(y\) changes based on the value of \(y\) itself. Because \(e^{y}\) is always greater than zero for real numbers \(y\), it impacts the landscape of potential solutions. For instance, when solving for constant solutions, we reasoned that \(e^{y} = 0\) cannot happen, which helps us conclude that \(t\) must equal zero instead of \(e^{y}\).

Understanding exponential functions involves recognizing their behavior in both logarithmic scales and real-world applications, such as population growth, radioactive decay, and certain financial models. Mastery of exponential functions opens doors to analyzing complex exponential growth or decay processes across disciplines.