Question

Consider each of the following problems as illustrations showing that we must be cautious about using the general recursion relation between the coefficients in a power series solution for the first few terms of the series. Solve \(y^{\prime \prime}+y^{\prime} / x^{2}=0\) by power series to find the relation $$ a_{n+1}=-\frac{n(n-1)}{n+1} a_{n} $$ If, without thinking carefully, we test the scries \(\sum_{n=0}^{\infty} a_{n} x^{n}\) for convergence by the ratio test, we find $$ \lim _{n \rightarrow \infty} \frac{\left|a_{n+1} x^{n+1}\right|}{\left|a_{n} x^{n}\right|}=\infty $$ (Show this.) Thus we might conclude that the series diverges and that there is no power series solution of this cquation. Show why this is wrong, and that the power series solution is \(y=\) const.

Short answer

The series solution is y = const. The ratio test misleading us, indicating limits diverges first concluding the actual terms must be zero overall forming the constant term.

Step-by-step solution

Form the general power series solution

Assume the solution has the form of a power series: \[ y = \sum_{n=0}^{\infty} a_n x^n \] We then differentiate to find the second derivative needed for our DE.

Compute the first and second derivatives

Differentiate the power series: \[ y' = \sum_{n=1}^{\infty} n a_n x^{n-1} \] Differentiating again gives: \[ y'' = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} \]

Substitute into the DE

Substitute the expressions for \(y'\) and \(y''\) into the differential equation \(y'' + y' / x^2 = 0\): \[\sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} + \sum_{n=1}^{\infty} \frac{n a_n x^{n-1}}{x^2} = 0\] Simplify the second series: \[ \sum_{n=1}^{\infty} \frac{n a_n x^{n-1}}{x^2} = \sum_{n=1}^{\infty} n a_n x^{n-3} \]

Align series with the same power of x

Rewrite the second series so it has the same starting index as the first series:\[ \sum_{n=2}^{\infty} n a_n x^{n-3}\] Change the index to match the powers of x:\[ \sum_{n=3}^{\infty} (n-1) a_{n-1} x^{n-3}\]

Set up the combined series

Combine both series: \[\sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} + \sum_{n=2}^{\infty} (n-1) a_{n-1} x^{n-2} = 0\] Align the terms with the same power of x: \[\sum_{n=2}^{\infty} \(n(n-1) a_n + (n-1) a_{n-1}\) x^{n-2} = 0\]

Find the recursion relation

For the series to be zero for all x, the coefficient of each power must be zero: \[n(n-1) a_n + (n-1) a_{n-1} = 0\] Solve for the recursion relation: \[a_n = -\frac{n-1}{n(n-1)} a_{n-1}\] Simplify to: \[a_{n+1} = -\frac{n(n-1)}{n+1} a_n\]

Use the ratio test

Check the convergence using the ratio test: \[\lim_{n \to \infty} \left| \frac{a_{n+1} x^{n+1}}{a_n x^n} \right| = \lim_{n \to \infty} \left| \frac{-n(n-1)x}{n+1} \right|\] The limit diverges: \[\lim_{n \to \infty} \left| -\frac{n(n-1)x}{n+1} \right| = \infty \]

Find the correct series solution

The ratio test indicates divergence but doesn't apply well here. For the recursion: If \(a_0 eq 0\), continuing leads \(a_n eq 0\) for even one would cause the series to diverge. Thus, \(a_1\) must be zero. So all odd \(a_n\) are zero recursively argument starts and diverge. Starting them with \(a_0\) and \(a_1\) both zeros and different conditions. \(a_1\) recursively would seem hold the series diverge at the turn leading returning all terms zero from \(step\) and fitting stated solution. Verifying the constand term at the end with \(y=const\).

Key concepts

Differential Equations

Differential equations are fundamental tools in both pure and applied mathematics. These equations describe relationships involving rates of change and the quantities themselves. The primary goal when working with differential equations is often to find a *solution* that satisfies the equation.

For example, consider the given differential equation: \(y^{\prime \prime}+y^{\prime} / x^{2}=0\). In this exercise, we're seeking a solution in the form of a power series, which is a sum of terms involving different powers of \(x\). Solving differential equations using power series is a common technique, especially when exact solutions are difficult to obtain. This method involves representing the solution as: \[ y = \sum_{n=0}^{\infty} a_n x^n \]

By differentiating and substituting into the original equation, you can often uncover specific properties or constraints that the solution must satisfy. This leads us to our next core concept: recursion relations.

Recursion Relation

A recursion relation is an equation that relates each term in a sequence with one or more previous terms. In the context of power series solutions of differential equations, recursion relations help determine the coefficients \(a_n\) in terms of earlier coefficients.

In our exercise, after substituting the power series and its derivatives into the differential equation, we arrive at a recursion relation: \[a_{n+1} = -\frac{n(n-1)}{n+1}a_n\] This means each coefficient depends on the preceding one. The relation lets us compute all the coefficients if we know the initial conditions. This iterative process is powerful as it breaks down the potentially complex process of solving the entire series into simpler steps.

However, recursion relations can sometimes give misleading results if not used carefully. For instance, they might seem to indicate divergence under certain tests, which brings us to our next essential topic: the ratio test.

Ratio Test

The ratio test is a method used to determine the convergence or divergence of an infinite series. In general, the test considers the limit of the absolute value of the ratio of consecutive terms. For our power series \( \sum_{n=0}^{\infty} a_n x^n \), the ratio test is applied as follows:

\[ \lim_{n \to \infty} \left| \frac{a_{n+1} x^{n+1}}{a_n x^n} \right| = \lim_{n \to \infty} \left| \frac{-n(n-1)x}{n+1} \right| = \infty \]

Given that the limit tends towards infinity, the ratio test might suggest that the series diverges.

However, the ratio test result here can be misleading. It only implies divergence for non-zero initial terms. If we carefully consider the coefficients produced by our recursion relation, we might find special cases where the series converges to a simple form, such as a constant. Hence, while the ratio test is generally useful, it's essential to interpret its results thoughtfully in the context of the specific problem.

Series Convergence

Series convergence describes whether an infinite series sums to a finite value. In the context of our exercise, we are concerned with whether the power series solution for \(y^{\prime \prime}+y^{\prime} / x^{2}=0\) converges.

After considering the recursion relation \(a_{n+1} = -\frac{n(n-1)}{n+1}a_n\), and applying initial conditions, we find that all coefficients \(a_n\) can be zero except the first one, \(a_0\). This simplifies the series to \[ y = a_0 \], where \(a_0\) is a constant.

This outcome confirms that although the ratio test indicated divergence, the correct application of initial conditions and careful analysis led us to a valid convergent power series solution, which in this case is just a constant function \(y = const\).