Question

Identify all the singular points of \(y^{\prime \prime}+p(t) y^{\prime}+q(t) y=0\) in the interval \(-10

Short answer

Answer: The singular point of the given differential equation is \(t = 0\).

Step-by-step solution

Analyze the expression \(1+\ln |t|\)

Check the range of the function \(1+\ln |t|\) for any values of \(t\) that make the function undefined. In this case, we are looking specifically for when the natural logarithm, \(\ln\), becomes undefined.

Determine when \(\ln |t|\) is undefined

The logarithm function is undefined when the argument is equal to zero. In this case, this means \(\ln |t|\) is undefined when \(|t| = 0\). But since, \(|t| ≥ 0\), it implies that \(\ln |t|\) is undefined when \(t = 0\).

Identify the singular points of the differential equation

Since we found that the natural logarithm is undefined at \(t=0\), the singular point of the given differential equation is: $$ t = 0 $$

Conclusion

The singular points of the given differential equation in the interval \(-10<t<10\) are: $$ t = 0 $$

Key concepts

Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives. They express how a function changes and are essential tools for modeling real-world phenomena such as physics, engineering, and biology.
Differential equations come in different types and forms. The main types include:

• Ordinary Differential Equations (ODEs): These involve functions of a single variable and their derivatives.
• Partial Differential Equations (PDEs): These involve functions of multiple variables and their partial derivatives.

In our exercise, we focus on an ordinary differential equation (ODE). To solve such an equation, we try to find the function (typically denoted as \(y\) in this case) that satisfies the relationship outlined by the equation.
Understanding how to handle differential equations allows us to predict how systems evolve over time under certain conditions.

Singularities

A singularity in a differential equation refers to the points where the equation "behaves badly," meaning that the equation or its solution might become undefined or non-differentiable. These points are crucial because they often indicate boundary conditions or places where the physical phenomena modeled by the equation can exhibit complex behavior.
In the context of this exercise, singularities occur when the coefficients of the differential equation become undefined. For the given equation, a singularity happens when \(1 + \ln |t|\) becomes undefined or starts behaving non-standardly, which is identified to occur at \(t = 0\). Finding singular points is a vital step in analyzing and solving differential equations as they dictate the domains where solutions are valid.

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is a logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm has unique properties that make it widely used in calculus, particularly in solving problems involving exponential growth and decay.
A central consideration with logarithms, especially when finding singularities, is understanding when they become undefined. The natural logarithm, \(\ln(x)\), is undefined for non-positive values since logarithms inherently express how many times to raise their base to obtain their argument. Therefore, \(\ln |t|\) becomes undefined precisely when \( |t| = 0 \), which only happens at \( t = 0 \).
The natural logarithm's properties mean it can simplify multiplicative processes, turning them into additive ones, making it a powerful tool in handling differential equations and other mathematical problems.

Boundary Value Problems

Boundary value problems (BVPs) are a type of differential equation where the solution is determined not just by the equation itself, but also by additional conditions specified at the boundaries of the domain. BVPs are critical for understanding physical systems where conditions are influenced by external constraints.
In many practical applications, the boundary conditions could include values such as points where the solution needs to satisfy certain behaviors, like zero displacement in a mechanical system or specific temperature at the edges in thermal problems. This differs from initial value problems, which specify conditions at a single point.
In our differential equation context, addressing singularities ensures we correctly understand where these boundary conditions can break down or be undefined, reinforcing the importance of ensuring those conditions make sense within the domain. For this exercise, knowing the singularity at \(t = 0\) informs us that any boundary value problems must be carefully checked to ensure they do not rely on values at the singular point.