← Differential Equations

Unit 10: Series Solutions

What to do when an equation like Airy's has no elementary solution: building one term by term as a power series instead.

8 topics · 28 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

5.1 Why Some Equations Need a Series Solution

  1. How it worksHow does assuming a solution is an infinite power series turn an unsolvable-looking equation into a solvable recursion?
  2. Common mistakeWhy doesn't the lack of elementary functions in a solution mean the differential equation has no real solution?
  3. Going furtherWhat happens to the series solution method once the equation's coefficients themselves aren't polynomials?
  4. Worked exampleFind the first three terms of the power series solution of y' = y about x = 0 and check that it matches ex.
Your turn1 of 2
Find the first three terms (through x2) of the power series solution of y' = 3y about x = 0, given y(0) = 4.

5.2 Series Solutions Near an Ordinary Point

  1. How it worksHow does matching coefficients of each power of x after substitution produce a recursion relation between the an's?
  2. Worked exampleFind the recursion relation and the first four nonzero terms of the series solution of y' - 2xy = 0 about x = 0.
  3. Worked exampleFind the first four terms of the series solution of y'' + y = 0 about x = 0 and compare them to the Taylor series of sine and cosine.
Your turn1 of 2
Find the first four nonzero terms (through x6) of the series solution of y' - 3xy = 0 about x = 0, given y(0) = 3.

5.3 Ordinary Points and Singular Points

  1. How it worksHow do you classify a point as ordinary, regular singular, or irregular singular by looking at p(x) and q(x)?
  2. Worked exampleFind whether x = 0 is an ordinary or singular point for x2 y'' + xy' + (x2 - 4)y = 0.
  3. Common mistakeWhy doesn't a singular point automatically mean the differential equation has no series solution there at all?
Your turn1 of 2
For the equation 2x2 y'' + 2xy' + (x2 - 6)y = 0, is x = 0 an ordinary point, a regular singular point, or an irregular singular point?

5.4 The Euler Equidimensional Equation

  1. How it worksHow does substituting y = xr turn the Euler equation into an algebraic indicial equation for r?
  2. Worked exampleSolve the Euler equation x2 y'' - 3xy' + 4y = 0 for x > 0.
  3. Going furtherWhat happens to the general solution of an Euler equation when its indicial equation has a repeated root?
  4. Worked exampleFind the general solution of the Euler equation x2 y'' + xy' - 4y = 0 for x > 0.
Your turn1 of 2
Solve the Euler equation x2 y'' + (1-4-0)xy' + 4·0y = 0 for x > 0, given y(1) = 4 and y'(1) = -1. Find y(x).

5.5 Regular Singular Points and the Method of Frobenius

  1. How it worksHow does the Frobenius method's indicial equation generalize the Euler equation's r2 term to a series case?
  2. Worked exampleFind the indicial equation and its roots for the regular singular point at x = 0 of 2x2 y'' + 3xy' - (1+x)y = 0.
  3. Going furtherWhat happens to the Frobenius method when the two roots of the indicial equation differ by an integer?
  4. Common mistakeWhy doesn't finding one Frobenius series solution automatically hand you the second independent one for free?
Your turn1 of 2
Find the indicial equation's roots for 2x2 y'' + 4xy' - (5+x)y = 0 at the regular singular point x=0: solve 2r(r-1)+4r-5=0 for the larger root.

5.5 Why the Frobenius Series Can Break Down

  1. Common mistakeWhy isn't it true that a series solution centered at an ordinary point only has to avoid singular points on the real axis?
  2. Going furtherWhat happens to a series solution's radius of convergence when the nearest singular point is a complex number rather than a real one?
  3. Ties togetherHow does the radius of convergence of a series solution connect to the radius of convergence of 1/(1+x2) as a Taylor series?
Your turn1 of 2
The function 1/(7+x2) has singular points at x = plus or minus i*√7 (complex, when a > 0). Find the radius of convergence of its Taylor series about x = 0.

5.3 Airy's Equation as a Series-Solution Example

  1. How it worksHow do the two independent series solutions of Airy's equation emerge from choosing y(0) = 1, y'(0) = 0 and the reverse choice?
  2. Worked exampleFind the first three nonzero terms of each of the two independent series solutions of y'' - xy = 0.
  3. Going furtherWhat happens to the coefficients of Airy's equation's series solution as you push further out in the recursion?
  4. Common mistakeWhy doesn't the ordinary point at x = 0 mean Airy's equation's series solution only converges near zero?
Your turn1 of 2
Find the first four terms (through x4) of the series solution of y'' - xy = 0 about x=0, given y(0) = 3 and y'(0) = 3.

5.1 Comparing a Series Solution to a Closed-Form Answer

  1. Worked exampleSolve y'' - y = 0 both by the characteristic equation method and by finding its power series solution about x = 0, and compare.
  2. Going furtherWhat happens to the series solution's terms when they're regrouped and recognized as the Taylor series of ex and e−x?
  3. Ties togetherHow does recognizing a series solution as a familiar Taylor series connect series methods back to constant-coefficient techniques?
Your turn1 of 2
Solve y'' - 32 y = 0 with y(0) = 3, y'(0) = 3, using the characteristic equation method. Find y(x).