Systems of ODEs — RK4 for systems — Maths notes

What we are solving (WHAT)

Deriving RK4-for-systems from scratch (HOW)

Start from the exact integral over one step: $\mathbf{y}(t+h) = \mathbf{y}(t) + \int_t^{t+h}\mathbf{f}(s,\mathbf{y}(s))\,ds.$

RK4 approximates this integral with Simpson-like sampling, evaluating $\mathbf f$ at the start, twice at the midpoint, and at the end — but using estimated states, not the unknown true ones.

Formula RK4 for a system (vector form)

One step $t_n \to t_{n+1}=t_n+h$ : $\mathbf{k}_1 = \mathbf{f}(t_n,\ \mathbf{y}_n)$ $\mathbf{k}_2 = \mathbf{f}\!\left(t_n+\tfrac{h}{2},\ \mathbf{y}_n+\tfrac{h}{2}\mathbf{k}_1\right)$ $\mathbf{k}_3 = \mathbf{f}\!\left(t_n+\tfrac{h}{2},\ \mathbf{y}_n+\tfrac{h}{2}\mathbf{k}_2\right)$ $\mathbf{k}_4 = \mathbf{f}\!\left(t_n+h,\ \mathbf{y}_n+h\,\mathbf{k}_3\right)$ $\boxed{\ \mathbf{y}_{n+1} = \mathbf{y}_n + \frac{h}{6}\big(\mathbf{k}_1 + 2\mathbf{k}_2 + 2\mathbf{k}_3 + \mathbf{k}_4\big)\ }$ Here each $\mathbf{k}_i$ is a vector of length $n$ . Local error $O(h^5)$ per step; global error $O(h^4)$ .

Figure — Systems of ODEs — RK4 for systems

Worked Example 1 — a 2×2 linear system

Solve $y_1' = y_2,\ \ y_2' = -y_1$ with $\mathbf y(0)=(1,0)$ . (True solution: $y_1=\cos t,\ y_2=-\sin t$ .) Take $h=0.1$ , one step.

So $\mathbf f(t,\mathbf y) = (y_2,\ -y_1)$ .

Step k1. $\mathbf k_1 = (y_2,\ -y_1) = (0,\ -1)$ . Why this step? Slope at the start using the current state $(1,0)$ .

Step k2. Intermediate state $\mathbf y + \tfrac h2\mathbf k_1 = (1,0)+0.05(0,-1)=(1,\ -0.05)$ . $\mathbf k_2 = (-0.05,\ -1)$ . Why this step? We re-evaluate the slope at the predicted midpoint — note both components used the advanced state, not just one.

Step k3. $\mathbf y + \tfrac h2\mathbf k_2 = (1,0)+0.05(-0.05,-1)=(0.9975,\ -0.05)$ . $\mathbf k_3 = (-0.05,\ -0.9975)$ . Why this step? A refined midpoint slope using $\mathbf k_2$ .

Step k4. $\mathbf y + h\,\mathbf k_3 = (1,0)+0.1(-0.05,-0.9975)=(0.995,\ -0.09975)$ . $\mathbf k_4 = (-0.09975,\ -0.995)$ . Why this step? Slope at the end of the interval.

Combine. $y_{1,1}=1+\tfrac{0.1}{6}\big(0+2(-0.05)+2(-0.05)+(-0.09975)\big)=1+\tfrac{0.1}{6}(-0.29975)\approx 0.99500.$ $y_{2,1}=0+\tfrac{0.1}{6}\big(-1+2(-1)+2(-0.9975)+(-0.995)\big)=\tfrac{0.1}{6}(-5.99)\approx -0.099833.$ True: $\cos 0.1=0.995004$ , $-\sin 0.1=-0.099833$ . The $y_1$ value matches to ~4 decimals; the $y_2$ value matches to ~3 decimals — with one step.

Worked Example 2 — a 2nd-order ODE via reduction

Solve $y'' + y = 0$ , $y(0)=1,\ y'(0)=0$ . Reduce: $u_1=y,\ u_2=y'$ : $u_1' = u_2,\qquad u_2' = -u_1.$ Why this step? Same physics as Example 1 — the pendulum/oscillator. We converted a 2nd-order ODE into a system so RK4-for-systems applies directly. The result is identical: $y_1$ tracks $\cos t$ . This proves the reduction trick lets one method handle all orders.

Forecast-then-Verify

80/20 — the few things that carry everything

Recall Feynman: explain to a 12-year-old

Imagine two friends running and holding hands — wherever one goes affects the other. To guess where they'll be in a moment, you don't peek once; you peek four times: at the start, twice in the middle, and at the end, each time guessing where both friends moved together. Then you take a clever weighted average (count the middle peeks double). Because they're holding hands, you must move both at each peek, never one alone. That careful four-peek average is RK4 for systems.

Connections

RK4 for a single ODE — the scalar parent method.
Reducing higher-order ODEs to first-order systems
Euler's method for systems — the cheaper, less accurate cousin.
Local vs Global Truncation Error — why $O(h^5)$ local → $O(h^4)$ global.
Stiff systems and stability — where explicit RK4 struggles.
Simpson's Rule — the quadrature RK4 mimics.

What change turns scalar RK4 into RK4 for a system?

Replace every scalar (

y

k_i

f

) by its vector version; recipe and weights are unchanged.

Why must you finish the full vector

\mathbf k_1

before starting

\mathbf k_2

Because each

f_i

depends on all components;

\mathbf k_2

needs the synchronized advanced state of every component.

How do you make a 2nd-order ODE

y''=g(t,y,y')

a first-order system?

Set

u_1=y,\ u_2=y'

; then

u_1'=u_2,\ u_2'=g(t,u_1,u_2)

RK4 final combination formula (vector)?

\mathbf y_{n+1}=\mathbf y_n+\frac h6(\mathbf k_1+2\mathbf k_2+2\mathbf k_3+\mathbf k_4)

Global error order of RK4?

O(h^4)

(local truncation

O(h^5)

\mathbf k_2

, what argument do you pass for the state?

\mathbf y_n+\tfrac h2\mathbf k_1

at time

t_n+\tfrac h2

What integral does RK4 approximate over one step?

\int_t^{t+h}\mathbf f(s,\mathbf y(s))\,ds

, sampled Simpson-style.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, ek single ODE mein sirf ek quantity ka rate hota hai. Lekin real life mein cheezein juddi hoti hain — jaise position aur velocity, ya predator aur prey. Inko alag-alag solve nahi kar sakte kyunki ek dusre par depend karte hain. Toh trick simple hai: saari unknowns ko ek vector $\mathbf{y}$ maan lo, aur jo RK4 scalar ke liye chalta tha, wahi recipe bilkul same — sirf har symbol ko bold (vector) bana do. Weights $1,2,2,1$ over $6$ same rehte hain.

Sabse important baat (yahi 80% marks dilata hai): stages ko synchronize karo. Pehle poora $\mathbf k_1$ nikalo — yaani saare components ka slope. Phir us advanced state se poora $\mathbf k_2$ , phir $\mathbf k_3$ , phir $\mathbf k_4$ . Galti yeh hoti hai ki log ek equation ka pura RK4 finish kar dete hain phir doosre ka — yeh galat hai kyunki $f_1$ ko $y_2$ ki updated value chahiye. Dono haath pakad ke chal rahe hain, ek ko akele aage nahi badha sakte.

Aur ek bonus: koi bhi higher-order ODE ko first-order system mein todo. $y''=-y$ ko $u_1=y,\ u_2=y'$ rakh ke $u_1'=u_2,\ u_2'=-u_1$ bana do. Ab same RK4-for-systems chala do. Isliye yeh ek hi method actually saari ODEs handle kar leta hai — bahut powerful aur exam mein guaranteed aata hai.

Systems of ODEs — RK4 for systems

What we are solving (WHAT)

Deriving RK4-for-systems from scratch (HOW)

Worked Example 1 — a 2×2 linear system

Worked Example 2 — a 2nd-order ODE via reduction

Forecast-then-Verify

80/20 — the few things that carry everything

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Numerical Methods

Connections