Differential equations
Differential equations
- A differential equation links a quantity to its rate of change.
- For a separable first-order equation: put all $y$ terms on one side, all $x$ terms on the other, then integrate both sides.
Practice
To solve a separable first-order differential equation, you:
Separate the variables, then integrate each side.
General vs particular
- The general solution contains a constant (a whole family of curves).
- An initial condition (a known value) fixes the constant → the particular solution.
- Example: $\dfrac{dy}{dx} = xy$ with $y = 1$ at $x = 0$ gives $y = e^{x^2/2}$.
Practice
An initial condition is used to:
A known value fixes the arbitrary constant, turning the general solution into the particular one.
Practice
The general solution of a differential equation contains an arbitrary constant.
Integrating introduces a constant, so the general solution is a whole family of curves.
You've got it
Key idea
- solve a separable equation by splitting variables and integrating both sides
- the general solution has a constant; an initial condition gives the particular solution
- a differential equation links a quantity to its rate of change