Differential Equations
Understand what a differential equation means, how to solve separable equations, and how initial conditions produce particular solutions.
1. What is a differential equation?
A differential equation is an equation involving a function and one or more of its derivatives. In modelling, derivatives often represent rates of change.
\[\frac{dy}{dx}=f(x,y)\]
A general solution contains a constant. A particular solution uses an initial condition such as \(y(0)=5\) to find that constant.
2. Separable example
Solve \(\frac{dy}{dx}=3x^2y\), where \(y\gt 0\).
Separate variables:
\[\frac{1}{y}\,dy=3x^2\,dx\]
Integrate both sides:
\[\ln y=x^3+C\]
Exponentiate:
\[y=Ae^{x^3}\]
where \(A=e^C\).
3. Numerical solutions: Euler's method
When an exact solution is difficult or unavailable, Euler's method uses small tangent-line steps.
\[y_{n+1}=y_n+h f(x_n,y_n)\]
The smaller the step size \(h\), the better the approximation usually becomes, but more steps are needed.