Question

Find all the regular singular points of the given differential equation. Determine the indicial equation and the exponents at the singularity for each regular singular point. \((x+1)^{2} y^{\prime \prime}+3\left(x^{2}-1\right) y^{\prime}+3 y=0\)

Short answer

Based on the given differential equation \((x+1)^{2} y^{\prime \prime}+3\left(x^{2}-1\right) y^{\prime}+3 y=0\), we found that there is a regular singular point at \(x = -1\). The exponents at this singularity are \(1\) and \(3\). These exponents are obtained from solving the indicial equation \((r - 1)(r - 3) = 0\).

Step-by-step solution

Identify the differential equation and rewrite it in the standard form (Euler equation)

Given differential equation is: \((x+1)^{2} y^{\prime \prime}+3\left(x^{2}-1\right) y^{\prime}+3 y=0\) We can rewrite this equation by dividing it by \((x+1)^2\) to obtain the standard form: \(y^{\prime \prime} + \frac{3(x^2-1)}{(x+1)^2} y^{\prime} + \frac{3}{(x+1)^2} y = 0\)

Determine the regular singular points

Regular singular points occur when the coefficient of \(y^{\prime}\) has simple poles. The coefficient in this case is: \(\frac{3(x^2-1)}{(x+1)^2}\) This expression has a simple pole at \(x = -1\). So, the only regular singular point is \(x = -1\).

Form the indicial equation

At the regular singular point, we look for a solution in the form \(y(x) = (x+1)^r\). Substitute this into the given differential equation to get the indicial equation in terms of \(r\): \([(r-1)r(x+1)^{r-2}]+3(x^2-1)(r(x+1)^{r-1})+3(x+1)^{r}=0\)

Solve the indicial equation to find the exponents

We look for a solution in the form \((x+1)^r\) with \(x = 0\). Plug this in the indicial equation: \((r-1)r+3(-1)r+3=0\) Simplify the equation: \(r(r-1)-3r+3=0\) \(r^2 - r - 3r + 3 = 0\) \(r^2 - 4r + 3 = 0\) Now factor the quadratic equation: \((r - 1)(r - 3) = 0\) So, we have two exponents: \(r_1 = 1\) and \(r_2 = 3\). To summarize, we found that the given differential equation has a regular singular point at \(x = -1\). The indicial equation is given by \((r - 1)(r - 3) = 0\). The exponents at the singularity for this regular singular point are \(1\) and \(3\).

Key concepts

Indicial Equation

An indicial equation is a fundamental tool used in the analysis of differential equations with regular singular points. When given a differential equation, such as \[(x+1)^{2} y^{\prime \prime}+3\left(x^{2}-1\right) y^{\prime}+3 y=0,\]we first convert it into a standard form by dividing through any terms to make it clearer, resulting in an equation like \[y^{\prime \prime} + \frac{3(x^2-1)}{(x+1)^2} y^{\prime} + \frac{3}{(x+1)^2} y = 0.\]
To identify an indicial equation, assume a solution of the form \(y(x) = (x+1)^r\), where \(r\) represents the roots of the indicial equation. Substitute this assumed solution back into the transformed differential equation. This substitution helps to focus on the behavior at the singular point, in this case, \(x = -1\).
You then equate terms to zero and solve for \(r\) to create the indicial equation. For the given example, the indicial equation is \[(r - 1)(r - 3) = 0,\]implying roots or exponents \(r = 1\) and \(r = 3\).

Exponents at Singularity

Exponents at a singularity are crucial because they inform us of the behavior of solutions around that point. In the context of differential equations, these exponents are derived from the indicial equation. In the previous example, the indicial equation \[(r - 1)(r - 3) = 0\]provided two exponents: \(r_1 = 1\) and \(r_2 = 3\). These numbers are known as the characteristic exponents of the solution.
Basically, they determine the order of the terms in a series solution around a regular singular point. These exponents play a role in forming a solution as part of a power series expansion.

• When the exponents are distinct: The solutions based on these are generally independent.
• When exponents differ by an integer: Special consideration might be necessary to find a second solution.

Knowing these exponents helps us understand potential growth or decay of solutions at the vicinity of singular points.

Differential Equation

A differential equation relates a function to its derivatives and can express a variety of physical phenomena. For example, the differential equation provided \[(x+1)^{2} y^{\prime \prime}+3\left(x^{2}-1\right) y^{\prime}+3 y=0\]shows the relationship involving the function \(y\) and its first and second derivatives with respect to \(x\).
Solving such equations often involves finding the function \(y(x)\) that satisfies the equation under given conditions, involving methods such as power series or transformations.Differential equations may contain singularities, points where certain solutions cease to exist or the equation loses its normal behavior. Here, we focus on regular singular points, which are manageable compared to irregular ones, using tools like the indicial equation to find solutions.
Understanding the foundational concepts of differential equations is essential for solving them effectively and applying them to real-world scenarios, such as engineering or physics problems.

Euler Equation

An Euler equation is a specific type of differential equation and can often be converted from the original form for simpler analysis. The given differential equation: \[(x+1)^{2} y^{\prime \prime}+3\left(x^{2}-1\right) y^{\prime}+3 y=0,\]was converted by dividing through by \((x+1)^2\) to standardize it:\[y^{\prime \prime} + \frac{3(x^2-1)}{(x+1)^2} y^{\prime} + \frac{3}{(x+1)^2} y = 0.\]Euler equations often have the form \[a(x) y^{\prime \prime} + b(x) y^{\prime} + c(x) y = 0,\]where solving involves finding solutions dependent on the power of the variable near singular points.
This normalization helps in identifying regular singular points, making it easier to apply methods such as Frobenius series to analyze the behavior of solutions. Euler's forms are particularly neat for seeking analytic solutions and set the stage for other techniques that require standard, predictable formats to solve effectively.