# Ordinary Differential Equations (ODEs): The Engineer's Handbook

> 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:**

1. Calculate the **Integrating Factor**: $I(x) = e^{\\int P(x) dx}$
    
2. Multiply the entire equation by $I(x)$.
    
3. The Left Hand Side (LHS) collapses via the Product Rule: ddx\[I(x)y\]=I(x)Q(x)
    
4. 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 $r\_1, r\_2$:

1. **Distinct Real Roots** ($b^2 - 4ac > 0$): y(t)=c1er1t+c2er2t
    
2. **Repeated Real Root** ($b^2 - 4ac = 0$): y(t)=c1ert+c2tert
    
3. **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:**

1. **Linearity:** $\\mathcal{L}{af + bg} = a\\mathcal{L}{f} + b\\mathcal{L}{g}$
    
2. **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
> 
> 1. Take $\\mathcal{L}$ of both sides of the ODE.
>     
> 2. Solve algebraically for $Y(s)$.
>     
> 3. Use Partial Fractions to decompose $Y(s)$.
>     
> 4. 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.