Intuition The one core ideaTo follow a curved path forward, don't trust the direction it points right now — peek at where it will point at the end of your step, then walk using the average of those two directions. Everything in Modified Euler (Heun's method) is just this "average two slopes" idea dressed in symbols.
This page assumes you have seen nothing . Before we can even read d x d y = f ( x , y ) after it has a plain-words meaning and a picture.
x y == x == == y == point ( x , y ) x y
Look at Figure 1. The horizontal chalk line is the x y path .
f ( x , y ) f machine : you feed it your current position x y slope — how steeply the path is rising or falling right where you stand. We write f ( 0 , 1 ) = 1 x = 0 1 1
Why does the topic need f ( x , y ) f ( x ) both where you are and how high you are. A hill's steepness at a spot may change depending on your altitude, so the machine needs both inputs.
Definition Slope — "rise over run"Slope answers: if I step a tiny bit to the right, how much do I go up? If moving right by 1 2 2 1 drops you by 0.4 − 0.4
Look at Figure 2. The little right triangle under the tangent line has a horizontal leg (the run ) and a vertical leg (the rise ). Slope = run rise
Intuition Why "slope" is the whole gameNumerical methods build a curve step by step , and the only information they have at each step is the slope. A big positive slope means "climb steeply next"; zero slope means "stay flat for now"; negative means "descend." If you can read a slope, you can predict the next tiny piece of the path.
Sign cases you must recognise:
Slope > 0
Slope = 0
Slope < 0
In Example 2 of the parent, the start slope was f ( 0 , 1 ) = 0 exactly zero , meaning the curve leaves the start flat before bending down. That is the "slope = 0
d x d y d d y d x d x d y rise-over-run shrunk down to a single point — the slope of the path exactly where you stand.
y ′ == y ′ == d x d y y y ′ = f ( x , y ) "the slope of the path equals whatever the rule f That single line is the problem the whole chapter solves.
Why does the topic need a special symbol for "slope at an instant"? Because a curve's slope changes from point to point. Rise-over-run over a big run gives an average; the derivative gives the slope at one exact place, which is what we need to aim our next step.
d x d y Why it feels right: it looks like b a The fix: treat it as one symbol meaning "instantaneous slope." d y d x
d x d y = f ( x , y ) at every point, my path's slope must equal the number the machine f It does not tell you the height directly — only the direction to head at each spot. Your job is to reconstruct the height everywhere from the slope rule.
Intuition Why one slope rule isn't enough — you need a starting heightInfinitely many paths have the same slope rule; they are parallel copies shifted up and down. To pin down one path you must be told a single anchor: one ( x , y ) initial condition .
y ( x 0 ) = y 0 == x 0 == == y 0 == y ( x 0 ) = y 0 x 0 y 0 x 0 = 0 y 0 = 1 ( 0 , 1 )
Look at Figure 3: many candidate curves share the slope rule (pale chalk), but only the one through the marked anchor point (yellow dot) is our solution.
h == h == x x 0 x 0 + h h h h = 0.1
x 1 y 1 == x 1 === x 0 + h == y 1 == y 1
∫ x 0 x 1 f d x ∫ sum . It adds up all the tiny rises (slope × x 0 x 1 total change in height across the step. This is why the parent writes
y ( x 1 ) − y ( x 0 ) = ∫ x 0 x 1 f ( x , y ) d x . (height at end) minus (height at start) = sum of all the little rises in between. That is exact — no approximation yet.
Why does the topic need an integral? Because the slope rule only gives instantaneous directions; to recover an actual height you must accumulate all those directions across the step. Accumulating tiny pieces is integration. The differences between Euler and Heun are just different shortcuts for estimating this one sum — see Euler's Method (left-rectangle) versus Trapezoidal Rule (average the two ends).
k 1 k 2 == k 1 == start : k 1 = f ( x 0 , y 0 ) == k 2 == predicted end : k 2 = f ( x 1 , y 0 + h k 1 ) f
y 1 ( p ) ( p ) predicted — a rough draft. y 1 ( p ) = y 0 + h k 1 k 2 y 1
Intuition Why two slopes and an averageThe start-slope k 1 k 1 k 2 2 1 ( k 1 + k 2 ) h
Derivative dy dx equals y prime
Slope rule dy dx equals f x y
Machine f x y outputs a slope
Initial condition y at x0 equals y0
Step size h and next point x1 y1
Two slopes k1 and k2 predictor
Heun method average of slopes
Test yourself — cover the right side.
What does x y x y
What does the machine f ( x , y ) A single number — the slope of the path at position x y
Define slope in plain words. Rise over run — how much you go up when you step a tiny bit to the right.
What does slope = 0 The path is momentarily flat (a peak, valley, or plateau).
What does y ′ d x d y They are the same thing — the slope of y
Read d x d y = f ( x , y ) The path's slope must equal whatever number the rule f
Why do we need the initial condition y ( x 0 ) = y 0 The slope rule alone allows infinitely many parallel curves; the anchor picks the one true path.
What are x 0 , y 0 , x 1 , h Start position, start height, next position = x 0 + h
What does ∫ x 0 x 1 f d x The total change in height over the step — the sum of all tiny rises.
What is y 1 ( p ) A cheap Euler guess of the end height, made only to read the end-slope; it is discarded after correcting.
What are k 1 k 2 The slope at the start and the slope at the predicted end; Heun walks using their average.
Euler's Method — builds the predictor step and the left-rectangle idea of §6.Trapezoidal Rule — the "average both ends" integral shortcut of §6–§7.Order of Accuracy and Step Size — what the step size h Predictor-Corrector Methods — where k 1 , k 2 , y 1 ( p ) Runge-Kutta Methods — Heun as the two-slope RK2.Taylor Series Methods — how the derivative expansions certify accuracy. 
