Question

Find the radius of convergence of a series solution about the origin for the equation \(\left(z^{2}+a z+b\right) y^{\prime \prime}+2 y=0\) in the following cases: (a) \(a=5, b=6 ;\) (b) \(a=5, b=7\). Show that if \(a\) and \(b\) are real and \(4 b>a^{2}\) then the radius of convergence is always given by \(b^{1 / 2}\).

Short answer

R = b^{1/2} provided \[4b > a^{2}\]

Step-by-step solution

- Identify singular points

To determine the radius of convergence, locate the singular points of the differential equation. Singular points occur when the coefficient of the highest derivative, \[z^{2} + az + b\], is zero.

- Solve for singular points

Solve the equation \[z^{2} + az + b = 0\] for \[z\]. This gives the possible values of \[z\] that might affect the radius of convergence.

- Cases (a) and (b)

(a) For \[a = 5, b = 6\]: Solve \[z^{2} + 5z + 6 = 0\]. The roots are \[z = -2 \] and \[z = -3\]. The distance to the nearest singular point from the origin is \[2\].(b) For \[a = 5, b = 7\]: Solve \[z^{2} + 5z + 7 = 0\]. The roots are complex, given by \[z = -\frac{5}{2} + \frac{i\frac{3}{2}}{2} \] and \[z = -\frac{5}{2} - \frac{3\frac{i}{2}}{2}\]. The modulus of these roots is \[ \frac{7^{1/2}}{2} \].

- Analyze radius of convergence formula

From the given information, check if \[4b > a^{2}\]. For both cases, verify if the conditions hold. Observe from the equation that the roots' modulus if \[4b > a^{2}\] is given by \[b^{1/2}\].

- Confirm the formula

In both cases, confirm that if \[4b > a^{2}\], the radius of convergence \[R\] is \[b^{1/2}\]. (a) For \[b = 6, a = 5\]: Calculate \[4 \times 6 > 25\] (False). Default to minimal impact root distance. (b) Case: Check for further root impact.

Key concepts

Singular Points

When solving a differential equation, identifying the singular points is crucial. Singular points occur where the coefficient of the highest derivative is zero. For the equation \( (z^2 + az + b) y'' + 2y = 0 \), we set the quadratic coefficient \(z^2 + az + b \) to zero and solve for \( z \). These values are the singular points of the equation. For example, with \( a = 5 \) and \( b = 6 \), solving \(z^2 + 5z + 6 = 0 \) gives the roots \(z = -2 \) and \(z = -3 \). These points are crucial when determining the radius of convergence for a series solution.

Differential Equations

A differential equation involves unknown functions and their derivatives. The goal is to find the function that satisfies the equation. In this context, we are particularly interested in linear differential equations with polynomial coefficients. For the given equation, \( (z^2 + az + b) y'' + 2y = 0 \), we observe that the term \(z^2 + az + b \) impacts the nature of solutions significantly. Breaking this down is essential in analyzing its behavior near singular points. To determine the radius of convergence for series solutions, we need to understand where this term equals zero, as these points (singular points) will influence where and how solutions can be expressed as a series.

Series Solutions

Series solutions involve expressing the solution of a differential equation as an infinite sum, typically a power series. The radius of convergence of this series is the distance from the expansion point (often the origin) to the nearest singular point of the equation. For example, to find the radius of convergence for \( (z^2 + az + b) y'' + 2y = 0 \), we first determine the singular points by solving \(z^2 + az + b = 0 \). The smallest absolute value among these solutions gives the distance to the nearest singular point, which is the radius of convergence. For \( a = 5 \) and \( b = 6 \), the singular points \(z = -2 \) and \(z = -3 \) give a convergence radius of \(2 \). In case \( b \) satisfies \(4b > a^2 \), the radius simplifies to \( \sqrt{b} \). This principle helps ensure that the series solution is valid within the determined radius.