Question

In each exercise, (a) Verify that \(t=0\) is a regular singular point. (b) Find the indicial equation. (c) Find the recurrence relation. (d) Find the first three nonzero terms of the series solution, for \(t>0\), corresponding to the larger root of the indicial equation. If there are fewer than three nonzero terms, give the corresponding exact solution. $$ t^{2} y^{\prime \prime}+t y^{\prime}+(t-9) y=0 $$

Short answer

If yes, find the first three nonzero terms of the series solution for \(t>0\) corresponding to the larger root of the indicial equation. Answer: Yes, \(t=0\) is a regular singular point for the given differential equation. The first three nonzero terms of the series solution for \(t>0\) corresponding to the larger root of the indicial equation are given by the exact solution \(y(t) = a_0\), where \(a_0\) is a nonzero constant.

Step-by-step solution

Verify that \(t=0\) is a regular singular point

To verify that \(t=0\) is a regular singular point, we will first check if the coefficients of the given differential equation are analytic at \(t=0\). We have: 1. \(t^2\) is analytic at \(t=0\) since it is a power of \(t\). 2. \(t\) is analytic at \(t=0\) since it is a power of \(t\). 3. \(t-9\) is also analytic at \(t=0\) since it is a linear function of \(t\). Therefore, all coefficients are analytic at \(t=0\), making it a regular singular point.

Find the indicial equation

Recall the Frobenius method for a series solution: $$y(t) = \sum_{n=0}^\infty a_n t^{n+r}$$ Differentiate \(y(t)\) with respect to \(t\) to get \(y'(t)\) and \(y''(t)\): $$y'(t) = \sum_{n=0}^\infty (n+r) a_n t^{n+r-1}$$ $$y''(t) = \sum_{n=0}^\infty (n+r)(n+r-1) a_n t^{n+r-2}$$ Now, substitute \(y\), \(y'\), and \(y''\) into the given differential equation: $$t^2\sum_{n=0}^\infty (n+r)(n+r-1) a_n t^{n+r-2} + t\sum_{n=0}^\infty (n+r) a_n t^{n+r-1} + (t-9)\sum_{n=0}^\infty a_n t^{n+r} = 0$$

Find the recurrence relation

Combine the three series in the equation and match the powers of \(t\): $$\sum_{n=0}^\infty [(n+r)(n+r-1) a_n t^{n+r} + (n+r) a_n t^{n+r} + a_n(t-9) t^{n+r}] = 0$$ The series will be zero for all \(t\) if the coefficients of each power of \(t\) are all zero: $$(n+r)(n+r-1) a_n + (n+r) a_n + a_n(t-9) = 0$$ This gives us the recurrence relation: $$a_{n+1} = \frac{(n+r)(n+r-1) + (n+r)}{(n+1)(n+r+1) (t-9)} a_n$$

Find the first three nonzero terms of the series solution for \(t > 0\)

To find the first three nonzero terms, we need to solve the indicial equation for \(r\). In this case, the indicial equation is given by: $$r(r-1) + r = 0$$ Solving for \(r\), we get \(r=0\) and \(r=-1\). The larger root is \(r=0\). Therefore, our series solution becomes: $$y(t) = \sum_{n=0}^\infty a_n t^n$$ Using the recurrence relation derived in Step 3, we can find the first three nonzero terms: 1. \(a_0\) is a nonzero constant as it is not determined by the recurrence relation. 2. \(a_1 = \frac{0}{1(t-9)} a_0 = 0\). 3. \(a_2 = \frac{1}{2(t-9)} a_1 = 0\). Our series solution corresponding to the larger root of the indicial equation is: $$y(t) = a_0$$ Since there are fewer than three nonzero terms, the corresponding exact solution is \(y(t) = a_0\), where \(a_0\) is a nonzero constant.

Key concepts

Regular Singular Point

In the analysis of differential equations, identifying the type of singular point we're dealing with is crucial for finding an appropriate solution. A regular singular point is a type of singular point where the equation doesn't behave normally, but with certain restrictions that allow for a more manageable solution process. To verify if a point is a regular singular point, we check if the coefficients of the lowest order derivatives of the differential equation have certain types of singularities.

For instance, in our exercise with the differential equation \(t^2 y''+t y'+(t-9) y=0\), we determined that \(t=0\) is a regular singular point since the coefficients are analyzable at \(t=0\). This means the Frobenius method can be used to find a series solution around that point. Recognizing regular singular points enables us to apply specialized solution techniques that wouldn't work for other kinds of singular points.

Indicial Equation

Solving a differential equation near a regular singular point involves finding a power series solution with terms of the form \(t^{n+r}\), where \(r\) is a number determined by what we call the indicial equation. The indicial equation originates from substituting the series solution into the differential equation and then finding the lowest power of \(t\) that appears (after canceling any common factors).

In our example, we derive the indicial equation from the leading coefficients of the series after substituting into the given differential equation and setting the series equal to zero. The solutions to the indicial equation, known as the roots, will influence the form of the series solution. They are essential as they tell us where to start our series expansion by giving us the initial exponent for \(t\).

Recurrence Relation

Once we have determined the indicial equation, the next step in using the Frobenius method is to establish the recurrence relation. The recurrence relation is a formula that relates each coefficient in the series to the previous one. In simpler terms, it's a sequential recipe that tells us how to find one 'ingredient' based on the one we prepared just before.

For the differential equation in our exercise, we saw how comparing coefficients resulted in a pattern we can use to calculate all subsequent terms once we have our starting values. These relations are invaluable because they often allow us to calculate many terms of the series solution with just a few simple calculations, essentially automating a portion of the solution process.

Frobenius Method

The Frobenius method is a powerful technique used to find series solutions near a regular singular point. It's based on the assumption that we can express the solution as a power series with coefficients that depend on the terms we're trying to find. This method is named after the mathematician Ferdinand Frobenius, who developed it.

When applying the Frobenius method, as demonstrated in our exercise, we start by assuming our series will have terms of the form \(a_n t^{n+r}\). We differentiate this series to get the necessary derivatives, and then substitute everything back into the original differential equation. Organizing the terms then leads us to the indicial equation and the recurrence relations, which give us a formal structure to find the coefficients of the series. In the end, if successful, the method provides us with a power series expression for the solution around our singular point.