= Isocline
{title2=$f(x,y)=k$}
For the first-order <ordinary differential equation> $y'=f(x,y)$ defined on an <open subset> $D$ of the plane, an <isocline> of <slope> $k$ is the <level set>
$$
\{(x,y)\in D:f(x,y)=k\}.
$$
Here $k$ is a constant. At every point of this <isocline>, the <direction field> prescribes a <tangent vector> proportional to $(1,k)$ for a <solution of a differential equation>. It does not generally prescribe the <tangent line> of the <isocline> itself. The <level set> can have several components or fail to be a smooth curve, and points outside $D$ are excluded.
For example, take $y'=x-y$ on the whole plane. Its <isoclines> are the parallel <straight lines>
$$
x-y=k,\qquad y=x-k.
$$
Each <line> has <slope> $1$, while the <direction field> along it has <slope> $k$. Thus the <isocline> $y=x$ carries horizontal segments even though the <line> itself rises. A <function graph> of a <solution of a differential equation> follows the segments, rather than following this <line>. Only the <isocline> with $k=1$, namely $y=x-1$, is also a <solution of a differential equation>.<18.03SC Differential Equations, Fall 2011/Direction Fields, Isoclines, Integral Curves>[pp. 1–2: constant-slope loci and the example y prime equals x minus y]{ref}
\Image[/isoclines-linear-direction-field.png]
{title=Isoclines label the prescribed slopes while exact solution graphs follow the direction field}
The <first-order linear differential equation> has the exact family
$$
y(x)=x-1+Ae^{-x},\qquad y'(x)=1-Ae^{-x}=x-y(x).
$$
The <initial condition> $y(x_0)=y_0$ selects $A=(y_0-x_0+1)e^{x_0}$ and hence a unique <solution of a differential equation> for every real $x$. The picture shows $y(0)=-1,0,1,2$. For $A=0$ the <solution of a differential equation> stays on the $k=1$ <isocline>; for other $A$ it moves through different <isoclines>. These formulas check the pictured trajectories by direct <differentiation>, independently of how the <direction field> was sampled.
The zero-slope <isocline> $f(x,y)=0$ is called a <nullcline>. Below the <nullcline> $y=x$ in this example, $y'>0$; above it, $y'<0$. For $A>0$, the <solution of a differential equation> meets it at $x=\log A$, where $y'=0$ and $y''=1$, a strict <local minimum>. The <nullcline> is therefore a useful guide to turning points, although it is not itself a <solution of a differential equation>. In the equivalent planar <autonomous differential equation> $\dot x=1$, $\dot y=x-y$, it is the <nullcline> for the second component; there is no <equilibrium point>, because the first component never vanishes.
This makes <isoclines> useful even when no explicit <solution of a differential equation> is available: organize a <direction field> by common <slopes>, read where <function graphs> rise or fall, and identify possible turning points on <nullclines>.
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