Mr. IntegralTHE WHOLE UNIT · ONE REFERENCE
Equations and Initial Values
Cheat sheet.
The key rules, formulas and reminders from all 3 topics, gathered into reference cards.
Key formulas, conditions and traps · Read down each column.
Differential Equations and Solution Families
Core rule
A solution satisfies the equation and all stated initial or boundary conditions on a specified interval.
Watch for
An algebraic expression without a valid interval is an incomplete answer. Nonlinear equations can have finite-time blow-up even when their formulas look elementary.
Initial Values, Existence and Uniqueness
Core rule
Continuous gives local existence; local Lipschitz continuity in gives uniqueness.
Watch for
For a linear equation , continuous coefficients on an interval give a unique solution throughout that interval. Divide by the coefficient of only where it is nonzero.
Direction Fields and Solution Curves
Core rule
At , the tangent slope is . Zero-slope curves need not themselves be solutions.
Watch for
Under uniqueness, distinct solution curves cannot cross at the same . A coarse slope field can hide rapid changes; numerical traces need a step-size check as well as a visually plausible curve.
Differential Equations and Solution Families
2 reference blocks
Core rule
A solution satisfies the equation and all stated initial or boundary conditions on a specified interval.
Watch for
An algebraic expression without a valid interval is an incomplete answer. Nonlinear equations can have finite-time blow-up even when their formulas look elementary.
Initial Values, Existence and Uniqueness
2 reference blocks
Core rule
Continuous gives local existence; local Lipschitz continuity in gives uniqueness.
Watch for
For a linear equation , continuous coefficients on an interval give a unique solution throughout that interval. Divide by the coefficient of only where it is nonzero.
Direction Fields and Solution Curves
2 reference blocks
Core rule
At , the tangent slope is . Zero-slope curves need not themselves be solutions.
Watch for
Under uniqueness, distinct solution curves cannot cross at the same . A coarse slope field can hide rapid changes; numerical traces need a step-size check as well as a visually plausible curve.