Question

Solve Airy's DE, \(y^{\prime \prime}+x y=0\), by the power-series method. Show that the radius of convergence for both independent solutions is infinite. Use the comparison theorem to show that for \(x>0\) these solutions have infinitely many zeros, but for \(x<0\) they can have at most one zero.

Short answer

The Airy's differential equation \(y''+x y=0\) possesses two independent power series solutions that both have an infinite radius of convergence. Furthermore, for \(x>0\), these solutions have an infinite number of zeros, and for \(x<0\), at most one zero.

Step-by-step solution

Defining the Power Series Method

Airy's differential equation, \(y''+xy=0\), may be solved by expressing \(y(x)\) as a power series: \(y(x) = \sum_{n=0}^{\infty} a_nx^n \). Then, we substitute this into the differential equation.

Calculating Coefficients

The result of substitution leads us to an expression that contains the sum of two series. By equating corresponding coefficients in these series we can derive recursive relations for the coefficients \(a_n\). The even and odd terms are derived separately, leading to two independent power series solutions.

Determining the Radius of Convergence

By utilizing the ratio test, it can be shown that the radius of convergence for both power series solutions is infinite. That is, the solution series converge for all real numbers \(x\). This is due to the fact that the coefficients \(a_n\) become smaller as \(n\) grows large.

Applying the Comparison Theorem

The comparison theorem states that if a power series solution \(y(x)\) for a differential equation has infinitely many zeros in an interval, then any other linearly independent solution must also have infinitely many zeros in this interval. Thus, since one solution of Airy's DE shows these properties for \(x>0\), it follows that another independent solution must as well. For \(x<0\), the solution can have at most one zero because the solution function has to remain monotonic (without maxima or minima).

Key concepts

Power Series Method

The power series method is a robust technique for finding solutions to differential equations. It involves assuming that the solution can be expressed as a series, typically a power series:
\( y(x) = \sum_{n=0}^{\infty} a_n x^n \).
By plugging this series representation into the differential equation, you transform a potentially difficult problem into one that involves understanding and manipulating the coefficients \( a_n \).
In Airy’s differential equation, this process helps us derive two sets of coefficients that correspond to linearly independent solutions. Changing variables or methods might not be possible, and therefore the power series is particularly useful in cases where standard analytical solutions do not exist. Additionally, from the series perspective, the problem boils down to solving a recurrence relation obtained by equating coefficients of like powers of \( x \). Each coefficient \( a_n \) depends on its predecessors, which allows systematic calculations.

Radius of Convergence

The radius of convergence is a fundamental concept in understanding where a power series converges to a function. Using the ratio test, a common tool in calculus, we can determine this radius. For the power series solution of Airy’s DE, the test involves examining the limit of the ratio of successive terms:
\[ L = \lim_{n \to \infty} \left| \frac{a_{n+1} x^{n+1}}{a_n x^n} \right| = \left| x \right| \cdot \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \].
In this case, it turns out that the radius of convergence is infinite because the coefficients \( a_n \) decrease rapidly, ensuring the series converges for all \( x \).
This implies the solutions to Airy’s DE are valid across the entire number line, offering great flexibility and insights into the solution behavior across different values of \( x \), whether positive or negative.

Comparison Theorem

The comparison theorem is a useful tool in examining the behavior of solutions to differential equations. For power series solutions, this theorem can help us understand the structure and number of zeros the solution might have on an interval.
According to the theorem, if a particular solution has infinitely many zeros in an interval, then any other linearly independent solution will also have infinitely many zeros within that same range. When applied to Airy’s DE for \( x > 0 \), this theorem affirms that the oscillatory nature of one solution will reflect in its independent pair.
However, for \( x < 0 \), conditions are quite different. The solutions might become non-oscillatory due to the dominance of certain terms in the power series expansion. This not only allows the possibility of having at most one zero but can fundamentally change our understanding of how solutions behave in negative domains where different physical or geometrical interpretations might arise.

Zeros of Solutions

Understanding the zeros of solutions is crucial in applying differential equations to real-world problems, as zeros often signify transitions or critical points.
For Airy’s DE, solutions show complex behavior distinguished by whether \( x \) is positive or negative.

• For \( x > 0 \): Solutions have infinitely many zeros. This oscillatory behavior is common in contexts like wave motion or alternating electrical currents.
• For \( x < 0 \): Solutions may have at most one zero. This indicates a more stable, monotonic-like behavior absent of repeated crossings of the x-axis, which can reflect stability in mechanical systems or growth/decay processes.

This underlines the importance of context when interpreting solutions. Decisions based on these zeros can impact engineering designs, stability analyses, and predictions of system behaviors.