ODE Sketch Lab · draw two curves · run a Butterworth low-pass · couple for ODE consistency

1) Sketch y(x) (left) and y′(x) (right).
2) Pick a filter + cutoff / order.   3) Toggle ODE coupling.
4) The spectra under each canvas tell you what the filter removed.
y(x) — your solution sketch before
points: 0 family C: — ‖y′ − dy/dx‖: —
y′(x) — claimed derivative before
points: 0 RHS: y ‖y′ − RHS‖: —

Branch picker — which solution do you keep?

your y (raw) your y′ (raw) filtered ȳ (FFT low-pass) filtered ȳ′ (FFT low-pass) branch solution Y(x; C)
Per-coord Butterworth runs on each sketch independently in the frequency domain (Y(k) ← H(k) · Y(k), H(k) derived from σ / cutoff / order). ODE coupling then iterates in the same FFT space: for each bin k, push P(k) toward jω·Y(k), and pull Y(k) back along jω. Strength α blends these into one operator.