4.1.1 · v1 · 70–90 min
First-order ODEs
Why this matters
ODEs describe how a quantity changes, connecting rates to trajectories in science, engineering and economics.
Learning objectives
- Identify independent/dependent variables, order, solution, general/particular solution, equilibrium and initial condition
- verify a proposed solution by substitution
- check units and domain.
Worked examples
Worked example 1
Given: y'=2y, y(0)=3. Goal: solve/verify. Method: exponential family. y=Ce^{2t}; initial condition gives C=3. Verification: y'=6e^{2t}=2y; y(0)=3. Interpretation: modeled quantity grows proportionally at rate 2 per time unit. Limit: constant rate and model scope.
Worked examples
Worked example 1
Given: T'=-.2(T-20), T(0)=80 °C. Solution: T=20+60e^{-.2t}. Verification: derivative -12e^{-.2t} equals -.2(60e^{-.2t}); initial value 80. Interpretation: approaches equilibrium 20°C without authorizing equipment operation.
Guided practice
- Determine order and variables in
x'=t−x. - verify
x=t−1+2e^{-t}by differentiation.
Common mistake and counterexample
Incorrect: treating y'=y as a function. Correction: solve for a function family and substitute; y=Ce^t.
Knowledge check and mastery evidence
four checks—roles/order, substitution, IC, units/domain; 4/4 required.
Related laboratory
ODE Solver.
Next lesson
Linear and separable equations.