Question

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

Short answer

Question: Identify the singular points of the given second-order differential equation \(y^{\prime \prime} + t^{2/3} y^{\prime} + (\sin t) y = 0\) in the interval \(-10 < t < 10\). Answer: The only singular point for the given differential equation in the interval \(-10 < t < 10\) is \(t = 0\).

Step-by-step solution

Identify singular points for the coefficient function \(p(t) = t^{2/3}\)

The function \(t^{2/3}\) is defined for all real numbers, but we need to check its differentiability. To do this, we can compute its first derivative: $$ \frac{d}{dt}t^{2/3} = \frac{2}{3}t^{-1/3} $$ The derivative is defined for all \(t \neq 0\). Therefore, \(t = 0\) is the only singular point for the coefficient function \(p(t)\) in the interval \(-10 < t < 10\).

Identify singular points for the coefficient function \(q(t) = \sin t\)

The function \(\sin t\) is defined and differentiable for all real numbers. Therefore, there are no singular points for the coefficient function \(q(t)\) in the interval \(-10 < t < 10\).

Combine singular points from both coefficients

Since the singular points for \(p(t) = t^{2/3}\) and \(q(t) = \sin t\) are disjoint, we can identify all the singular points of the given differential equation by taking the union of their singular points. Therefore, the only singular point for the given differential equation \(y^{\prime \prime} + t^{2/3} y^{\prime} + (\sin t) y = 0\) in the interval \(-10 < t < 10\) is: $$ t = 0 $$

Key concepts

Differential Equations

Differential equations are mathematical equations that involve unknown functions and their derivatives. They play a crucial role in modeling real-world phenomena, such as physics, engineering, and economics. In this example, we are dealing with a second-order linear differential equation:

\[y'' + p(t) y' + q(t) y = 0\]

The equation relates a function \(y(t)\) to its first and second derivatives, \(y'(t)\) and \(y''(t)\). Understanding how solutions behave, especially when coefficient functions like \(p(t)\) and \(q(t)\) are involved, is key.

• "Second-order" means the highest derivative is \(y''\).
• "Linear" implies the function and its derivatives appear to the first power, with no multiplication between them.

Differential equations can sometimes have singular points, which are specific values of \(t\) where behaviors change, making them crucial for analysis.

Coefficient Functions

Coefficient functions, denoted by \(p(t)\) and \(q(t)\) in our equation, influence how the solution behaves. They appear as multipliers of the derivatives and the function itself. In the given differential equation, these are expressed as:

• \(p(t) = t^{2/3}\)
• \(q(t) = \sin(t)\)

The function \(p(t) = t^{2/3}\) poses potential challenges, as its differentiability affects the equation's behavior at \(t = 0\). This makes \(t = 0\) a singular point for \(p(t)\). Conversely, \(\sin(t)\) is well-behaved, with no singular points in the interval \(-10 < t < 10\).

Coefficient functions must be carefully analyzed to identify any singular behaviors or points that might affect the equation’s solutions. This ensures accurate modeling and understanding of the system being studied.

Interval Analysis

Interval analysis involves examining how functions behave within a specific range. For differential equations, it helps determine where singular points or specific behaviors might occur.

In our exercise, the interval given is \(-10 < t < 10\). We analyze the behavior of the coefficient functions \(p(t) = t^{2/3}\) and \(q(t) = \sin(t)\) within this range.

• \(t^{2/3}\) is not differentiable at \(t = 0\), making it a singular point within the interval.
• \(\sin(t)\) has no such issues, remaining smooth and continuous throughout the entire interval.

Through interval analysis, we identify potential disruptions in the solution's behavior and ensure the reliability of the model we are examining.

Differentiability

Differentiability is a key concept in calculus, indicating that a function has a derivative at each point within its domain. It's essential for analyzing smoothness and continuity in solutions to differential equations.

In this problem, differentiability is crucial for examining \(p(t) = t^{2/3}\). Derivatives reveal where a function might behave unpredictably. For \(t^{2/3}\), the derivative \(\frac{2}{3}t^{-1/3}\) is undefined at \(t = 0\). This makes \(t = 0\) a singular point.

• Singular points occur where derivatives are undefined or discontinuities appear.
• A differentiable function is smooth, allowing for predictable solutions.

Recognizing where differentiability breaks down helps us understand why certain points are singular and how they affect the broader differential equation.