Series Solutions About an Ordinary Point
When the coefficients are not constant, assume a power series, match coefficients and generate a recurrence. Several named functions of physics are defined this way and no other.
What a learner can do afterwards
- Substitute a power series and derive the recurrence relation
- Generate the first several coefficients from initial conditions
- Say what the radius of convergence means for the solution's range
1 · Read
When the coefficients are not constant, assume a power series and let it tell you its own terms. Write y as the sum of a sub n times x to the n, differentiate term by term, shift the indices so like powers line up, and match coefficients. That matching hands you a recurrence relation linking each coefficient to the earlier ones. Several named functions of physics are defined this way and no other.
Solve y prime equals 3 y with a0 equals 2. Term by term differentiation turns the equation into n plus 1 times a sub n plus 1 equals 3 a sub n. At n equals 0 that reads 1 times a1 equals 3 times 2, so a1 is 6. At n equals 1 it reads 2 times a2 equals 3 times 6, so a2 is 9. Feed each answer into the next step and the series builds itself: 2 plus 6 x plus 9 x squared onward.
Check your young series against friends you already know. The series above is 2 times e to the 3 x expanded, and matching a known expansion catches algebra slips fast. Also mind where the series can be trusted: the radius of convergence says how far from the centre the sum still represents the solution. This chapter works at ordinary points, where that machinery runs smoothly.
Reproduce each algebra step on paper with the indices written out fully. Most mistakes hide in shifted indices and lost factorials, and slow index bookkeeping beats clever shortcuts. Grow pattern sense too: denominators collecting factorials mean you are on track.
Substitute the series, shift indices, read the recurrence, and grow the coefficients step by step.
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8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.