General form
Damping and forcing may both vary in time.
du/dt + a(t)u = b(t)
General First-Order Linear ODEs
Solving linear dynamics through integrating factors
An integrating factor converts a linear ODE into an exact derivative and separates transient memory from forced response.
watchone idea
→manipulateone example
→leave withone intuition
The essential idea: multiplying by the right time-dependent factor makes the unknown appear inside one derivative.
One Concept · One Example
General First-Order Linear ODEs
Presented by Charlotte Moser · Coming soon
What to notice
- Linearity excludes powers such as u².
- The homogeneous part carries initial-condition memory.
- The forcing builds the particular response.
The mathematical structure
Constant coefficients
For fixed a and b, the transient decays and the forced response approaches b/a.
u(t)=e⁻ᵃᵗu₀+(b/a)(1−e⁻ᵃᵗ)
Separate memory from response
Use the constant-coefficient case from the slides and reveal how its two terms combine.
solution u(t)equilibrium b/a
Integrating factors expose the transient and forced pieces of linear dynamics.