Differential Equations (ODE) Reference

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.

• $y^{\prime }+y=0$ (First Order)
• $y^{\prime \prime }+3y^{\prime }+2y=0$ (Second Order)

Linearity: An ODE is linear if the dependent variable $y$ and its derivatives $y^{\prime },y^{\prime \prime }$ appear to the first power and are not multiplied together.

• Linear: $y^{\prime \prime }+\sin (t)y=e^t$
• Non-Linear: $y^{\prime \prime }+y^2=0$ (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.

$$\frac{dy}{dx}=g(x)h(y)⟹\int \frac{1}{h(y)}dy=\int g(x)dx$$

B. Linear Equations (Integrating Factor)

Standard form:

$$\frac{dy}{dx}+P(x)y=Q(x)$$

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: $$\frac{d}{dx}[I(x)y]=I(x)Q(x)$$
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:

$$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$$

Solution: $F(x,y)=C$.

---

3. Second Order Linear ODEs ($n=2$)

Typically used to model springs, circuits, and vibrations. Standard Form:

$$a{y}^{\prime \prime }+b{y}^{\prime }+cy=f(t)$$

Part 1: Homogeneous ($f(t)=0$)

We assume the solution is of the form $y=e^{rt}$. This leads to the Characteristic Equation:

$$a{r}^2+br+c=0$$

Solve for roots $r_1,r_2$:

1. Distinct Real Roots ($b^2-4ac>0$): $y(t)=c_1{e}^{r_{1}t}+c_2{e}^{r_{2}t}$
2. Repeated Real Root ($b^2-4ac=0$): $y(t)=c_1{e}^{rt}+c_{2}t{e}^{rt}$
3. Complex Roots ($r=\lambda \pm \mu i$): $y(t)=e^{\lambda t}(c_1\cos (\mu t)+c_2\sin (\mu t))$

Part 2: Non-Homogeneous ($f(t)\ne 0$)

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 $y_p(t)$
$A{e}^{kt}$	$C{e}^{kt}$
$\sin (kt)$ or $\cos (kt)$	$A\cos (kt)+B\sin (kt)$
Polynomial $t^n$	$A_n{t}^n+\cdots +A_0$

---

4. The Laplace Transform ($\mathcal{L}$)

The engineer’s “Hack”. It turns Calculus problems (derivatives) into Algebra problems.

Definition:

$$F(s)=\mathcal{L}\{f(t)\}={\int }_0^{\infty }{e}^{-st}f(t)dt$$

Key Properties:

1. Linearity: $\mathcal{L}\{af+bg\}=a\mathcal{L}\{f\}+b\mathcal{L}\{g\}$
2. Differentiation: Turns derivatives into multiplication by $s$. $\mathcal{L}\{y^{\prime }(t)\}=sY(s)-y(0)$ $\mathcal{L}\{y^{\prime \prime }(t)\}=s^{2}Y(s)-sy(0)-y^{\prime }(0)$

Common Transforms Table:

$f(t)$	$F(s)$
$1$	$1/s$
$e^{at}$	$1/(s-a)$
$\sin (kt)$	$k/(s^2+k^2)$
$\cos (kt)$	$s/(s^2+k^2)$
$t^n$	$n!/s^{n+1}$

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.

$$y_{n+1}=y_n+h\cdot f(t_n,y_n)$$

• $h$: 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.

$$y_{n+1}=y_n+\frac{h}{6}(k_1+2k_2+2k_3+k_4)$$

Where:

• $k_1=f(t_n,y_n)$
• $k_2=f(t_n+\frac{h}{2},y_n+\frac{h}{2}{k}_1)$
• $k_3=f(t_n+\frac{h}{2},y_n+\frac{h}{2}{k}_2)$
• $k_4=f(t_n+h,y_n+h{k}_3)$

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.