Mr. IntegralTHE WHOLE UNIT · ONE REFERENCE
Series, Boundary Values and Checkpoint
Cheat sheet.
The key rules, formulas and reminders from all 4 topics, gathered into reference cards.
Key formulas, conditions and traps · Read down each column.
Power-Series Solutions
Core rule
Shift all sums to the same power before equating coefficients; use initial data to start the recurrence.
Watch for
A truncated series is an approximation and needs an error or convergence argument for its intended interval. Singular points may require a Frobenius series rather than ordinary nonnegative integer powers; that is a further topic beyond this introductory method.
Fourier Series and Periodic Forcing
Core rule
Compute harmonic coefficients, apply each frequency response, then superpose where convergence justifies it.
Watch for
A finite Fourier sum overshoots near a jump; adding terms narrows the affected region but does not eliminate the limiting Gibbs overshoot. For an undamped oscillator, a resonant harmonic needs separate treatment because no bounded steady periodic response exists.
Boundary-Value Problems and Eigenvalues
Core rule
Watch for
In engineering, eigenfunctions describe spatial mode shapes; time evolution enters through an associated dynamic model. Orthogonality of distinct sine modes helps decompose a shape or forcing. Do not confuse the free amplitude of an eigenfunction with a unique boundary-value solution.
Differential Equations: Engineering Checkpoint
Core rule
Model → classify → solve → verify data and residual → inspect stability and units.
Watch for
The expanded course supplies a foundation, not every special-function or partial-differential-equation method. Use the checkpoint to identify gaps: a wrong initial slope suggests missing homogeneous terms; an unexpected jump suggests a forcing error; alternating growth in a decaying model suggests numerical instability.
Power-Series Solutions
2 reference blocks
Core rule
Shift all sums to the same power before equating coefficients; use initial data to start the recurrence.
Watch for
A truncated series is an approximation and needs an error or convergence argument for its intended interval. Singular points may require a Frobenius series rather than ordinary nonnegative integer powers; that is a further topic beyond this introductory method.
Fourier Series and Periodic Forcing
2 reference blocks
Core rule
Compute harmonic coefficients, apply each frequency response, then superpose where convergence justifies it.
Watch for
A finite Fourier sum overshoots near a jump; adding terms narrows the affected region but does not eliminate the limiting Gibbs overshoot. For an undamped oscillator, a resonant harmonic needs separate treatment because no bounded steady periodic response exists.
Boundary-Value Problems and Eigenvalues
2 reference blocks
Core rule
Watch for
In engineering, eigenfunctions describe spatial mode shapes; time evolution enters through an associated dynamic model. Orthogonality of distinct sine modes helps decompose a shape or forcing. Do not confuse the free amplitude of an eigenfunction with a unique boundary-value solution.
Differential Equations: Engineering Checkpoint
2 reference blocks
Core rule
Model → classify → solve → verify data and residual → inspect stability and units.
Watch for
The expanded course supplies a foundation, not every special-function or partial-differential-equation method. Use the checkpoint to identify gaps: a wrong initial slope suggests missing homogeneous terms; an unexpected jump suggests a forcing error; alternating growth in a decaying model suggests numerical instability.