Question

Find all singular points of the given equation and determine whether each one is regular or irregular. \(x^{2}(1-x) y^{\prime \prime}+(x-2) y^{\prime}-3 x y=0\)

Short answer

Question: What are the singular points of the given equation \(x^2(1-x)y''+(x-2)y' -3xy = 0\) and determine if they are regular or irregular. Answer: Both singular points \(x = 0\) and \(x = 1\) are regular singular points.

Step-by-step solution

Identify the equation's singular points

To find the singular points, we first look at the leading coefficient of the highest order derivative, which is \(x^2(1-x)\). A singular point occurs when this leading coefficient is zero. So, we solve the equation: \(x^2(1-x) = 0\) By solving this equation, we obtain the singular points \(x = 0\) and \(x = 1\).

Find the indicial equation

To determine the nature of each singular point, we need to find the corresponding indicial equation. To do that, we can divide the given equation by the leading coefficient of the highest order derivative: \(\frac{x^2(1-x)y''+(x-2)y' -3xy}{x^2(1-x)} = 0\) By simplifying, we get the equation: \(y'' + \frac{(x-2)}{(1-x)}y' - \frac{3}{(1-x)}y = 0\) Now, we can assume a solution in the form of a power series: \(y(x) = \sum_{n=0}^\infty a_n x^{r+n}\), where \(r\) is to be determined. We then compute the derivative (\(y'(x)\)) and the second derivative (\(y''(x)\)): \(y'(x) = \sum_{n=0}^\infty (r+n) a_n x^{r+n-1}\) \(y''(x) = \sum_{n=0}^\infty (r+n)(r+n-1) a_n x^{r+n-2}\) Next, we substitute the series for \(y(x)\), \(y'(x)\), and \(y''(x)\) back into the simplified equation: \(\sum_{n=0}^\infty (r+n)(r+n-1) a_n x^{r+n-2} + \frac{(x-2)}{(1-x)} \sum_{n=0}^\infty (r+n) a_n x^{r+n-1} - \frac{3}{(1-x)} \sum_{n=0}^\infty a_n x^{r+n} = 0\)

Compute indicial equation

To find the indicial equation, we need to evaluate the coefficient of the lowest power of \(x\) for each singular point. For \(x=0\), we look at the coefficient of \(x^{r-2}\) in the first term and \(x^{r-1}\) in the second term. The lowest powers are \(r = 2\) and \(r = 1\) respectively, so the indicial equation must have \(r = 2\). For \(x=1\), we look at the coefficient of \(x^{r+1-2}\) in the first term and \(x^{r-1}\) in the second term. The lowest powers are \(r = 1\) and \(r = 2\) respectively, so the indicial equation must have \(r = 1\).

Determine the nature of singular points

To determine if the singular points are regular or irregular, we must check the root difference of the indicial equation. For \(x = 0\), since \(r = 2\), the root difference is \(2 - 2 = 0\), so the singular point \(x = 0\) is regular. For \(x = 1\), since \(r = 1\), the root difference is \(1 - 1 = 0\), so the singular point \(x = 1\) is regular. In conclusion, both singular points \(x = 0\) and \(x = 1\) are regular singular points.

Key concepts

Singular Points

In the realm of differential equations, understanding singular points is essential. Singular points are specific values of the variable, often x, where a differential equation's leading coefficient becomes zero. These points are crucial because they can significantly affect the behavior of solutions to the differential equation.

In the given exercise, we identify the singular points by setting the leading coefficient of the highest-order derivative to zero. Here, the leading coefficient is \(x^2(1-x)\). Solving \(x^2(1-x) = 0\) gives us the singular points at \(x = 0\) and \(x = 1\). These are the potential problem areas where the behavior of the equation might change or become undefined. Recognizing these points helps us analyze whether the solutions are regular or need a special kind of approach.

Regular and Irregular Singularities

Once singular points are identified, the next step is to determine whether they are regular or irregular. Regular singular points allow the solution to be expressed around them using simpler methods. This occurs when, in a particular transformed version of the equation, no terms become undefined or infinite.

Irregular singularities, on the other hand, are more complex and may cause solutions to exhibit abrupt changes or even become infinite. In our exercise, to check the nature of the singular points, each one must be evaluated using its indicial equations. If the indicial equation has a consistent or zero root difference, then the singular point is regular. The solution to these can be determined using more straightforward series methods, unlike irregular singularities which often require more advanced techniques. In this example, both \(x = 0\) and \(x = 1\) were found to be regular because their root differences were zero.

Indicial Equation

The indicial equation is a crucial concept when examining singular points and their nature. It helps in determining the kind of solution that can be expected around these points. Essentially, it comes into play when we assume a power series solution and need to find an equation for the characteristic exponent, \(r\).

The indicial equation is derived by examining the lowest power of \(x\) in the expansion of a series solution substitution into the differential equation. For each singular point, this process is repeated to evaluate and simplify the related equations. The resulting equation indicates the behavior and characteristics of solutions near singularities, suggesting how the solution can be constructed and whether it remains valid around these points.

Power Series Solutions

Power series solutions are a valuable tool in solving differential equations, especially around singular points. The method involves assuming that the solution can be expressed as an infinite sum of powers of the variable, typically in the form \(y(x) = \sum_{n=0}^\infty a_n x^{r+n}\).

This series approach allows one to systematically find a solution that satisfies the differential equation at or near a singular point. In the example exercise, this method was used to explore the behavior around singular points \(x = 0\) and \(x = 1\). By substituting this series into the differential equation and simplifiying, one can determine the coefficients \(a_n\) in the series, which describe the solution. This method, particularly reliable for regular singular points, gives a structured way to handling complex equations that may not yield to simple algebraic manipulations.