Example 2 — The Plane \( f(x,y) = y - x \)

A plane and its level lines

The plane \( z = y - x \) with level lines \( y - x = C \)

Given the ODE

\( y' = y - x, \)

the surface \( f(x,y) = y - x \) describes a plane.

The level curves of this plane are the lines

\( y - x = C \;\Rightarrow\; y = x + C. \)

Vectors of constant slope on each isocline

On each isocline of height \( C \) we draw vectors whose slope is \( C \):

For \( C = 2 \) we get \( y = x + 2 \). On this line we draw vectors with direction \( \arctan(2) \approx 63^\circ \).

For \( C = 1 \) we get \( y = x + 1 \). On this line we draw vectors with direction \( \arctan(1) = 45^\circ \).

For \( C = 0 \) we get \( y = x \). On this line we draw vectors with direction \( \arctan(0) = 0^\circ \).

For \( C = -1 \) we get \( y = x - 1 \). On this line we draw vectors with direction \( \arctan(-1) = -45^\circ \).

Line isoclines of \( y' = y-x \): every vector on the same isocline points in the same direction

The dense direction field and three solutions

The dense direction field of \( y' = y-x \) and the solutions for \( y(0)=0 \), \( y(0)=2 \), \( y(2)=1 \)

Continuing this way we obtain the dense direction field. Note that the vectors of the same isocline all point in the same direction. That direction is \( \arctan C \), where \( C \) is the height of the isocline on the surface \( f(x,y) = y - x \).

Every solution of the ODE must advance so that it is always tangent to the direction field. The panel shows the solutions obtained for

\( y(0) = 0, \qquad y(0) = 2, \qquad y(2) = 1. \)
© Shir Sivroni