Differential equations involving first derivatives; model cooling, charging and decay.
A first-order ODE relates y and y'. Examples: y'=-ky (decay), y'+ay=b (charging). Solutions contain one arbitrary constant fixed by an initial condition. They model cooling, RC and RL circuits, mixing and population growth.
Newton cooling dT/dt=-k(T-Tₐ) gives T=Tₐ+(T₀-Tₐ)e^{-kt}.
First-order ODEs model any quantity that changes at a rate proportional to its present state, such as cooling, charging and decay.
Differential equations involving first derivatives; model cooling, charging and decay.
First-order ODEs model any quantity that changes at a rate proportional to its present state, such as cooling, charging and decay.
It is written dy/dt equals f of t and y, and it needs one initial condition to define a unique solution.
Many processes are driven by a rate proportional to the current state, such as Newton's law of cooling, so a single lag or storage element gives this form.
Typical applications include predicting the temperature of a cooling part over time, estimating the voltage across a capacitor as it charges and modelling the level of liquid in a tank with a constant outflow.
Study "Singular value decomposition" first (topic 58 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
Direction fields show behaviour. Equilibrium solutions where y'=0. Time constant τ=1/k.
Newton cooling dT/dt=-k(T-Tₐ) gives T=Tₐ+(T₀-Tₐ)e^{-kt}.
The next topic is "Separable equations", which builds directly on this one.
It is classed as Intermediate in Engineering Mathematics (topic 59 of 104). It is an intermediate topic. Work through the beginner topics in Engineering Mathematics first.
Want to practise? Open the interactive course for quick checks, quizzes, simulations and progress tracking.