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Series solutions · Differential Equations | Math Infinitum

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Differential Equations
Series solutions

Differential Equations · Chapter 08

Series solutions

Equations whose coefficients vary have no characteristic equation, and their solutions are often no combination of familiar functions. A power series still finds them: the equation fixes the coefficients one by one, the singular points say how far the series reaches, and a power of x in front rescues the method at the mildest singular points.

Equations whose coefficients vary have no characteristic equation, and their solutions are often no combination of familiar functions. A power series still finds them: the equation fixes the coefficients one by one, the singular points say how far the series reaches, and a power of x in front rescues the method at the mildest singular points.

y = n = 0 \sum \infty a_{n} (x - x_{0})^{n}

y^{'} = n = 1 \sum \infty n a_{n} x^{n - 1}

a_{n + 2}

∣ x - x_{0} ∣ < R

x^{r} n = 0 \sum \infty a_{n} x^{n}

y (x_{0}) = a_{0}

r (r - 1)

P (x) y^{''} + Q (x) y^{'} + R (x) y = 0

Lessons

Build the foundation

01Why series solutionsThe methods so far need constant coefficients, or a lucky substitution. Many equations of physics have variable coefficients — Airy’s equation for light near a rainbow’s edge, Legendre’s and Bessel’s equations for spheres and drums — and their solutions are not built from exponentials, sines or polynomials. They can still be found as power series: guess that the solution is an infinite polynomial, and let the equation decide its coefficients one by one.Read lesson →
02The power series methodSubstitute the series into the equation, shift the indices so that every sum is written in the same power of x, and collect: the result is one power series that must be zero, so every one of its coefficients must be zero. Each of those conditions is an equation linking a few coefficients of the solution. Worked through for an equation whose answer is already known, the method finds that answer, which is the best evidence that it works.Read lesson →
03Working out the coefficientsWith variable coefficients the method is the same, and now it earns its keep. Each term of the equation contributes a sum; after shifting, the coefficient of each power gives one equation, and solving it for the coefficient with the highest index gives that coefficient from earlier ones. The coefficients split into two families, one started by the initial value and one by the initial slope, and each family is one solution.Read lesson →
04Recurrence relationsThe recurrence is the whole solution in compressed form. Unwound, it gives a formula for every coefficient, and from that formula two things can be read: whether the series stops, which makes the solution a polynomial, and how far from the starting point the series converges. The ratio of consecutive coefficients answers the second question, just as the ratio test does in the analysis course.

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05Series and initial value problemsAn initial value problem fixes the first two coefficients: the value at the starting point is the constant term and the slope is the coefficient of the first power. Every other coefficient then follows from the recurrence, so the solution is determined term by term. When the starting point is not zero, the series is written in powers of the distance from it, and the same steps apply after a change of variable.Read lesson →

06Ordinary and singular pointsWhere does the method work, and how far does the series reach? Put the equation in standard form, with the coefficient of the second derivative equal to one. A point where both remaining coefficients are convergent power series is an ordinary point, and there the method always works; a point where one of them blows up is singular. The series about an ordinary point converges at least out to the nearest singular point — even a singular point that is a complex number, off the real line.Read lesson →

07The Frobenius methodAt a singular point a power series can fail, but at the mildest kind — a regular singular point, such as the origin for Bessel’s equation — a small change rescues the method. Multiply the series by a power of x whose exponent r is to be found: the lowest power in the equation then gives a quadratic for r, the indicial equation, and each root starts a series. The exponent need not be a whole number, so a solution can behave like the square root of x near the singular point, or like a negative power.Read lesson →

08Chapter practiceBring the chapter together: substitute a series and shift its indices, find and unwind a recurrence, fix the first coefficients from initial values, locate the singular points and the guaranteed reach, and find the exponents at a regular singular point. No worked example sits above the answers.Read lesson →