Question

In each exercise, find the singular points (if any) and classify them as regular or irregular. $$ \left(t^{2}-1\right) y^{\prime \prime}+(t-1) y^{\prime}+y=0 $$

Short answer

Question: Determine the singular points (if any) and classify them as regular or irregular for the given differential equation: \( \left(t^{2}-1\right) y^{\prime \prime}+(t-1) y^{\prime}+y=0 \). Answer: The given differential equation has two singular points, \(t=1\) and \(t=-1\), both of which are regular singular points.

Step-by-step solution

Identify the differential equation and its coefficients

The given differential equation is: $$ \left(t^{2}-1\right) y^{\prime \prime}+(t-1) y^{\prime}+y=0 $$ This is a second-order linear differential equation. The coefficients are: $$ P(t) = \left(t^{2}-1\right) \quad \text{for} \quad y^{\prime \prime} $$ $$ Q(t) = (t-1) \quad \text{for} \quad y^{\prime} $$ $$ R(t) = 1 \quad \text{for} \quad y $$

Find the singular points

Singular points occur when the coefficient of the highest order derivative becomes zero or undefined. In this case, we focus on the coefficient P(t) for the term \(y^{\prime \prime}\): $$ P(t) = \left(t^{2}-1\right) = (t-1)(t+1) $$ P(t) becomes zero when \(t=1\) or \(t=-1\). These are our singular points.

Classify the singular points as regular or irregular

We can classify the singular points as regular or irregular by looking at the coefficients around the singular points. For the singular point \(t=1\), we analyze the functions \(P'(t)\) and \(Q(t)\): $$ P'(t) = \frac{d}{dt} (t-1)(t+1) = 2t $$ $$ P'(1) = \lim_{t \to 1} 2t = \lim_{t \to 1} 2 = 2 $$ Also, \(Q(t) = (t-1)\), so: $$ Q(1) = (1-1) = 0 $$ Since both \(P'(1)\) and \(Q(1)\) are finite, the singular point \(t=1\) is a regular singular point. For the singular point \(t=-1\), we analyze the functions \(P'(t)\) and \(Q(t)\): $$ P'(-1) = \lim_{t \to -1} 2t = \lim_{t \to -1} -2 = -2 $$ Also, \(Q(t) = (t-1)\), so: $$ Q(-1) = (-1-1) = -2 $$ Since both \(P'(-1)\) and \(Q(-1)\) are finite, the singular point \(t=-1\) is also a regular singular point.

Conclusion

In conclusion, the given differential equation has two singular points, \(t=1\) and \(t=-1\), both of which are regular singular points.

Key concepts

Second-Order Linear Differential Equation

A second-order linear differential equation is a type of differential equation that involves the second derivative of a function. These equations are crucial in modeling various physical systems, such as mechanical vibrations and electrical circuits. The general form of a second-order linear differential equation is:\[ a(t) y'' + b(t) y' + c(t) y = 0 \]Here, \( y'' \) represents the second derivative of \( y \) with respect to \( t \). The functions \( a(t) \), \( b(t) \), and \( c(t) \) are the coefficients that may vary with \( t \). In solving these equations, one often looks for a function, \( y(t) \), that meets both the equation itself and any initial or boundary conditions. Understanding the nature of these coefficients is essential, particularly when considering singular points, which can complicate or enrich the solution process.

Regular Singular Point

Regular singular points are significant in differential equations. They occur when certain conditions are met while analyzing the coefficients of the differential equation. To classify a singular point as regular, we need the point at which the coefficient for the highest-order term becomes zero or goes to infinity to meet specific criteria.

• The coefficient \( P(t) \) of the second derivative \( y'' \) must be zero at this point, which defines it as a singular point.
• The limit of \( \frac{(t - t_0)Q(t)}{P(t)} \) as \( t \rightarrow t_0 \) must exist and be finite.
• The limit of \( \frac{(t - t_0)^2 R(t)}{P(t)} \) as \( t \rightarrow t_0 \) should also exist and be finite.

If these conditions are satisfied, then the point is classified as a regular singular point. Regular singular points allow for solutions called Frobenius series, which are very powerful in solving differential equations near these points.

Irregular Singular Point

Irregular singular points are less straightforward compared to regular ones. They occur when the criteria that define regular singular points are not met. Specifically, if the limits described for a regular singular point either do not exist or are infinite, the point is classified as an irregular singular point.

• This usually occurs because the behavior of the coefficients makes it impossible to ensure that both limits are finite.
• Irregular singular points often indicate more complex behavior of the solutions near these points, which may involve growth rates that aren't manageable by typical power series methods, like the Frobenius method.

Understanding irregular singular points is essential because they can suggest unusual or challenging behaviors in the system modeled by the differential equation, requiring specialized techniques to explore solutions around these points.

Coefficient Analysis

Analyzing the coefficients of a differential equation is a crucial part of understanding its singular points, whether they are regular or irregular. Coefficient analysis involves examining how these coefficients interact at various points along the domain of the solution.For a differential equation of the form: \[ (t^2 - 1) y'' + (t - 1) y' + y = 0 \]we focus on:

• \( P(t) = t^2 - 1 \), which determines where the second derivative term becomes zero or undefined. Solving \( P(t) = 0 \) gives the singular points \( t = 1 \) and \( t = -1 \).
• The linear coefficient \( Q(t) = t - 1 \), analyzed at singular points for further classification.
• The constant term \( R(t) = 1 \), which helps determine the nature of behavior at these points.

Through this analysis, one can classify and understand the nature of singular points, which dictates the search for solutions and the methods used to find them. These insights are valuable for both theoretical understanding and practical applications of differential equations.