Mr. IntegralTHE WHOLE UNIT · ONE REFERENCE
First-Order Methods
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.
Separable Equations and Lost Equilibria
Core rule
Integrate only after recording zeros of .
Watch for
Absolute values in logarithms allow either sign before initial data select a branch. Integration constants cannot repair a lost equilibrium after an invalid division at that equilibrium.
First-Order Linear Equations
Core rule
Watch for
The complementary part carries initial-condition memory; the forced part reflects input. Their long-time behavior depends on the coefficient sign. A positive constant damping coefficient erases initial differences exponentially.
Exact Equations and Potential Curves
Core rule
Watch for
Multiplying by a factor that vanishes or blows up can change which points are admissible. State the working domain and check excluded points separately. A level curve may have a vertical tangent and fail to be one global graph.
Bernoulli Equations and Substitutions
Core rule
Watch for
The inverse substitution can introduce branch restrictions or a finite-time singularity. A linear transformed solution need not correspond to a real original solution on its entire interval.
Equilibria, Stability and Logistic Growth
Core rule
For , draw sign arrows between every equilibrium before interpreting stability.
Watch for
Carrying capacity is a modeling assumption, not a universal population law. Negative populations are mathematically possible in some equations but usually outside the model’s intended domain.
Cooling, Mixing and RC Circuits
Core rule
Watch for
Newton cooling uses the same shape with temperature difference from a fixed ambient value. If volume, ambient temperature or input varies, revise the coefficients or forcing instead of reusing a constant-parameter formula.
Separable Equations and Lost Equilibria
2 reference blocks
Core rule
Integrate only after recording zeros of .
Watch for
Absolute values in logarithms allow either sign before initial data select a branch. Integration constants cannot repair a lost equilibrium after an invalid division at that equilibrium.
First-Order Linear Equations
2 reference blocks
Core rule
Watch for
The complementary part carries initial-condition memory; the forced part reflects input. Their long-time behavior depends on the coefficient sign. A positive constant damping coefficient erases initial differences exponentially.
Exact Equations and Potential Curves
2 reference blocks
Core rule
Watch for
Multiplying by a factor that vanishes or blows up can change which points are admissible. State the working domain and check excluded points separately. A level curve may have a vertical tangent and fail to be one global graph.
Bernoulli Equations and Substitutions
2 reference blocks
Core rule
Watch for
The inverse substitution can introduce branch restrictions or a finite-time singularity. A linear transformed solution need not correspond to a real original solution on its entire interval.
Equilibria, Stability and Logistic Growth
2 reference blocks
Core rule
For , draw sign arrows between every equilibrium before interpreting stability.
Watch for
Carrying capacity is a modeling assumption, not a universal population law. Negative populations are mathematically possible in some equations but usually outside the model’s intended domain.
Cooling, Mixing and RC Circuits
2 reference blocks
Core rule
Watch for
Newton cooling uses the same shape with temperature difference from a fixed ambient value. If volume, ambient temperature or input varies, revise the coefficients or forcing instead of reusing a constant-parameter formula.