Mr. IntegralTHE WHOLE UNIT · ONE REFERENCE
Second-Order Linear Equations
Cheat sheet.
The key rules, formulas and reminders from all 6 topics, gathered into reference cards.
Key formulas, conditions and traps · Read down each column.
Superposition and Fundamental Solutions
Core rule
Watch for
Superposition applies to homogeneous linear equations. Adding two solutions with the same nonzero forcing doubles that forcing. Nonlinear equations generally do not allow superposition at all.
Characteristic Roots and Free Response
Core rule
Use the root type to select an independent real basis before applying initial conditions.
Watch for
The sign of the real parts controls exponential growth or decay. Purely imaginary roots produce bounded oscillation in this simple second-order homogeneous case, but do not attract neighboring solutions.
Undetermined Coefficients and Resonance
Core rule
When a forcing trial overlaps the homogeneous basis, multiply by enough powers of .
Watch for
Resonance in this ideal undamped model produces unbounded growth under sustained forcing. Real damping changes the response. A large finite peak in a damped system is different from the exact secular growth shown here.
Damping and Mechanical Transients
Core rule
Watch for
Increasing damping beyond critical can slow the dominant return mode. Claims about “fastest settling” depend on initial data and a chosen settling criterion; distinguish that design question from the root classification.
Variation of Parameters
Core rule
Watch for
Changing integration constants only adds homogeneous terms, so one convenient set is enough for a particular solution. Definite integrals from an initial time are useful for enforcing zero initial response and for numerical evaluation.
Frequency Response and RLC Models
Core rule
Watch for
The forcing frequency of the largest displacement amplitude need not equal . For the standard model a nonzero peak occurs at only when . Undamped resonance needs a separate time-domain solution.
Superposition and Fundamental Solutions
2 reference blocks
Core rule
Watch for
Superposition applies to homogeneous linear equations. Adding two solutions with the same nonzero forcing doubles that forcing. Nonlinear equations generally do not allow superposition at all.
Characteristic Roots and Free Response
2 reference blocks
Core rule
Use the root type to select an independent real basis before applying initial conditions.
Watch for
The sign of the real parts controls exponential growth or decay. Purely imaginary roots produce bounded oscillation in this simple second-order homogeneous case, but do not attract neighboring solutions.
Undetermined Coefficients and Resonance
2 reference blocks
Core rule
When a forcing trial overlaps the homogeneous basis, multiply by enough powers of .
Watch for
Resonance in this ideal undamped model produces unbounded growth under sustained forcing. Real damping changes the response. A large finite peak in a damped system is different from the exact secular growth shown here.
Damping and Mechanical Transients
2 reference blocks
Core rule
Watch for
Increasing damping beyond critical can slow the dominant return mode. Claims about “fastest settling” depend on initial data and a chosen settling criterion; distinguish that design question from the root classification.
Variation of Parameters
2 reference blocks
Core rule
Watch for
Changing integration constants only adds homogeneous terms, so one convenient set is enough for a particular solution. Definite integrals from an initial time are useful for enforcing zero initial response and for numerical evaluation.
Frequency Response and RLC Models
2 reference blocks
Core rule
Watch for
The forcing frequency of the largest displacement amplitude need not equal . For the standard model a nonzero peak occurs at only when . Undamped resonance needs a separate time-domain solution.