Lesson 1 of 9 · 6 min
What a differential equation is
NCERT §9.1–9.2
The science club keeps a change log. Nobody writes down where things are; they write down how fast they are changing: the chai on the staff-room table, a savings deposit, a culture in the lab.
The story this chapter follows: The science club's change log
The lesson in notes
In short
An ordinary equation such as x² − 5x + 6 = 0 is solved by numbers. A differential equation contains the derivative of an unknown function, for example dy/dx = 3y, and is solved by a function.
Definition: an equation that involves one or more derivatives of a dependent variable with respect to an independent variable is a differential equation.
If all the derivatives are with respect to a single independent variable, the equation is ordinary. Equations with partial derivatives in several variables are partial differential equations; this chapter deals only with ordinary ones.
Notation: y′ = dy/dx, y″ = d²y/dx², y‴ = d³y/dx³; for higher orders write yₙ for dⁿy/dxⁿ.
Finding an antiderivative is the simplest case: dy/dx = 2x is a differential equation whose solutions are y = x² + C.
Rates of change appear across science: the speed of a cooling drink, the growth of money at compound interest and the decay of a radioactive sample are all described by differential equations.
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Why equations of change matter, with animations
3Blue1Brown · English · Lecture · Open on YouTube