Linear Isoclines — the Plane \( f(x,y) = y - x \)

The slope surface

For

\( y' = y - x \)

the slope function is the plane \( f(x,y) = y - x \).

The slope surface — The plane \( z = y - x \)

Three isoclines

Three isoclines — Isoclines \( y=x+C \) with their constant-slope arrows

The isoclines \( y - x = C \) are parallel lines \( y = x + C \). On each line every direction segment has the same slope \( C \).

Building the skeleton

Drawing the isoclines one after another builds the skeleton of the direction field.

Building the skeleton — Isoclines appearing one by one

The full field

The full field — The field assembling isocline by isocline

Many isoclines fill the plane with direction segments — the direction field.

Solutions

Solution curves run tangent to the field everywhere. Note the exceptional straight-line solution \( y = x + 1 \).

Solutions — Solution curves on the field

Linear Isoclines — the Plane f(x,y) = y - x — Further solutions of \( y'=y-x \)

More solutions, from initial conditions on both sides of the line \( y = x + 1 \).