Ordinary Differential Equations (ODEs): The Engineer's Handbook
A concise guide to classifying, solving, and approximating the equations that describe change over time.
The Language of Change A Differential Equation (DE) describes how a state changes over time.
Algebra solves for unknown numbers: $x^2 + 2x = 0$
Differential Equations solve for unknown functions: $\frac{dy}{dx} = ky$
In Computer Science, we rarely solve these analytically (by hand). We solve them numerically using algorithms like Euler's Method or Runge-Kutta.
1. Classification & Definitions
Order: The highest derivative present in the equation.
$y' + y = 0$ (First Order)
$y'' + 3y' + 2y = 0$ (Second Order)
Linearity: An ODE is linear if the dependent variable $y$ and its derivatives $y', y''$ appear to the first power and are not multiplied together.
Linear: $y'' + \sin(t)y = e^t$
Non-Linear: $y'' + y^2 = 0$ (Because of $y^2$)
2. First Order ODEs ($n=1$)
A. Separable Equations
The simplest form. We move all $y$'s to one side and all $x$'s to the other. dydx=g(x)h(y)⟹∫1h(y)dy=∫g(x)dx
B. Linear Equations (Integrating Factor)
Standard form: dydx+P(x)y=Q(x)
Algorithm:
Calculate the Integrating Factor: $I(x) = e^{\int P(x) dx}$
Multiply the entire equation by $I(x)$.
The Left Hand Side (LHS) collapses via the Product Rule: ddx[I(x)y]=I(x)Q(x)
Integrate both sides and solve for $y$.
C. Exact Equations
If $M(x, y)dx + N(x, y)dy = 0$ is "exact", then there exists a potential function $F(x,y)$ such that: ∂M∂y=∂N∂x Solution: $F(x, y) = C$.
3. Second Order Linear ODEs ($n=2$)
Typically used to model springs, circuits, and vibrations. Standard Form: ay″+by′+cy=f(t)
Part 1: Homogeneous ($f(t) = 0$)
We assume the solution is of the form $y = e^{rt}$. This leads to the Characteristic Equation: ar2+br+c=0
Solve for roots $r_1, r_2$:
Distinct Real Roots ($b^2 - 4ac > 0$): y(t)=c1er1t+c2er2t
Repeated Real Root ($b^2 - 4ac = 0$): y(t)=c1ert+c2tert
Complex Roots ($r = \lambda \pm \mu i$): y(t)=eλt(c1cos(μt)+c2sin(μt))
Part 2: Non-Homogeneous ($f(t) \neq 0$)
The general solution is $y(t) = y_h(t) + y_p(t)$.
$y_h$: The homogeneous solution (from Part 1).
$y_p$: The "Particular" solution.
Method of Undetermined Coefficients: Guess the form of $y_p$ based on $f(t)$:
| Term in $f(t)$ | Guess for $y_p(t)$ |
| $Ae^{kt}$ | $Ce^{kt}$ |
| $\sin(kt)$ or $\cos(kt)$ | $A\cos(kt) + B\sin(kt)$ |
| Polynomial $t^n$ | $A_n t^n + \dots + A_0$ |
4. The Laplace Transform ($\mathcal{L}$)
The engineer's "Hack". It turns Calculus problems (derivatives) into Algebra problems.
Definition: F(s)=Lf(t)=∫0∞e−stf(t)dt
Key Properties:
Linearity: $\mathcal{L}{af + bg} = a\mathcal{L}{f} + b\mathcal{L}{g}$
Differentiation: Turns derivatives into multiplication by $s$. Ly′(t)=sY(s)−y(0) Ly″(t)=s2Y(s)−sy(0)−y′(0)
Common Transforms Table:
| $f(t)$ | $F(s)$ |
| $1$ | $1/s$ |
| $e^{at}$ | $1/(s-a)$ |
| $\sin(kt)$ | $k/(s^2 + k^2)$ |
| $\cos(kt)$ | $s/(s^2 + k^2)$ |
| $t^n$ | $n! / s^{n+1}$ |
[!tip] Solving IVPs with Laplace
Take $\mathcal{L}$ of both sides of the ODE.
Solve algebraically for $Y(s)$.
Use Partial Fractions to decompose $Y(s)$.
Take the Inverse Laplace $\mathcal{L}^{-1}$ to find $y(t)$.
5. Numerical Methods (CS Approach) 💻
When an ODE is impossible to solve by hand (which is 99% of real-world cases), we approximate.
Euler's Method
The simplest approach. We use the tangent line to step forward in time. yn+1=yn+h⋅f(tn,yn)
$h$: Step size (e.g., 0.01).
Error: $O(h)$ (Global error is proportional to step size).
Runge-Kutta 4 (RK4)
The industry standard for simulations (Game physics, orbital mechanics). It samples the slope at 4 points to predict the next step.
yn+1=yn+h6(k1+2k2+2k3+k4) Where:
$k_1 = f(t_n, y_n)$
$k_2 = f(t_n + \frac{h}{2}, y_n + \frac{h}{2}k_1)$
$k_3 = f(t_n + \frac{h}{2}, y_n + \frac{h}{2}k_2)$
$k_4 = f(t_n + h, y_n + hk_3)$
Error: $O(h^4)$. Much more precise than Euler.
Linked Notes
[[CPP-Ultimate-Guide]] - Implementing RK4 in C++.
[[Linear-Algebra]] - Eigenvalues for Systems of ODEs.
[[Physics-Simulation]] - Using ODEs for game engines.
