One Slope Field, Different Starting Values
A differential equation such as \(y'=F(x,y)\) assigns a slope to each point \((x,y)\) in the plane. Its slope field is shared by every solution: each solution curve follows the direction indicated at its points. But curves that start at different points can follow different paths through that same field.
To compare two such curves, first compare their initial slopes. Then ask how their vertical separation changes as \(x\) changes. Different slopes can make the gap grow or shrink; equal slopes at one point do not mean the curves are the same. Whether distinct paths can actually meet depends on a condition that ensures solutions to an initial-value problem are unique.
Compare the Slopes at the Starting Points
The differential equation gives the tangent slope at every point on a solution. In particular, if \(p(x_0)=y_1\) and \(q(x_0)=y_2\), then
These values describe the curves’ local directions. A positive slope means the solution is increasing at that point; a negative slope means it is decreasing; and a zero slope means its tangent is horizontal there. If the slopes differ, the curves initially point in different directions. If the slopes are equal, their tangent lines are parallel at their respective starting points, but the curves can still have different values and different paths.
Worked Example: Read Two Initial Directions
Consider \(y'=x-y\). Compare solutions starting at \((0,0)\) and \((0,2)\).
At \((0,0)\), the differential equation gives the slope
So the first solution has a horizontal tangent at its initial point. At \((0,2)\), its slope is
So the second solution is decreasing at its initial point, with tangent slope \(-2\). The local linear approximations, written from the point-slope formula, are \(y\approx 0+0(x-0)=0\) for the first curve and \(y\approx 2-2(x-0)=2-2x\) for the second.
Check: Substituting each starting point into \(x-y\) gives exactly the slope used: \(0-0=0\) and \(0-2=-2\). These calculations determine the initial directions, not the entire solution curves.
Equal Slopes Do Not Mean Equal Curves
If the right-hand side does not depend on \(y\), then all solutions have the same slope at a given \(x\). They may still have different vertical positions. For example, consider \(y'=x\). The curves \(p(x)=\frac{x^2}{2}-1\) and \(q(x)=\frac{x^2}{2}+2\) both satisfy the equation, but they start at different values.
Worked Example: Curves with the Same Slope at Each \(x\)
For \(p(x)=\frac{x^2}{2}-1\), differentiate and substitute:
Thus \(p'=F(x,p(x))\). For \(q(x)=\frac{x^2}{2}+2\),
Thus \(q\) also satisfies the same differential equation. At \(x=0\), their initial values are \(p(0)=-1\) and \(q(0)=2\), and both slopes are \(0\). At any \(x\), their signed separation is
Their separation remains \(3\), even though their slopes match at every common \(x\). Check: Differentiating each displayed curve gives \(x\), as required, and subtracting their values gives \(3\) at every \(x\).
This example also shows why a slope field describes directions rather than locations. At a fixed \(x\), the two curves encounter field marks with the same slope, but they occupy different heights. In a general equation \(y'=F(x,y)\), the slopes at those heights may differ, so the vertical gap may change.
What Can Happen to the Separation?
For solutions \(p\) and \(q\), define their signed separation by \(r(x)=q(x)-p(x)\). Wherever both curves are differentiable, subtracting their equations gives
This identity explains how the field affects the gap. If \(r'(x)>0\), the signed separation is increasing at that point; if \(r'(x)<0\), it is decreasing. A changing gap does not by itself mean the curves are crossing. If \(r\) remains positive, for instance, the curve \(q\) stays above \(p\) even if the gap becomes smaller.
A useful sufficient condition for distinct solution curves not to meet is that \(F\) be locally Lipschitz in \(y\). On a rectangle, this means there is a constant \(L\ge 0\) such that, for points in the rectangle,
Under the usual local existence assumptions—\(F\) continuous in \(x\) and \(y\), and locally Lipschitz in \(y\)—solutions of the same initial-value problem are locally unique. The following argument shows why the Lipschitz condition gives uniqueness on an interval where the solutions remain in such a rectangle.
Why Uniqueness Prevents a Meeting
For a complete local argument, suppose \(p\) and \(q\) are solutions on a compact interval \([a,b]\), have \(p(a)=q(a)\), and remain in a region where one Lipschitz constant \(L\) works. Set \(r=p-q\). Since both curves satisfy the equation,
Because \(r\) is continuous on \([a,b]\), it is bounded there; let \(M=\max_{a\le t\le b}|r(t)|\). Then \(|r'(t)|\le LM\) throughout the interval. The Mean Value Theorem implies \(|r(t)-r(s)|\le LM|t-s|\), so \(r\) is Lipschitz and therefore absolutely continuous. The absolutely continuous version of the Fundamental Theorem of Calculus now gives the integral identity used below.
Since \(r(a)=0\), absolute continuity gives \(r(t)=\int_a^t r'(s)\,ds\).
The Lipschitz condition implies \(|r(t)|\le L\int_a^t |r(s)|\,ds\).
Using \(|r(s)|\le M\) first, then applying the integral inequality repeatedly, gives \(|r(t)|\le M\frac{L^n(t-a)^n}{n!}\) for every positive integer \(n\).
For fixed \(t\), \(\frac{L^n(t-a)^n}{n!}\) tends to \(0\) as \(n\) grows. Hence \(|r(t)|=0\), so \(p(t)=q(t)\).
The same argument applies in the direction of decreasing \(x\), by reversing the interval. Thus, if two solutions meet at some \(c\), they solve the same initial-value problem with initial value at \(c\). Uniqueness forces them to agree throughout their common interval. In particular, two solutions that start with different values cannot meet on that interval. If one starts below the other, their order remains the same; the distance between them need not stay constant.
When the Nonintersection Conclusion Can Fail
The Lipschitz condition matters. Without it, distinct solutions can start at different values and meet later. Consider the equation \(y'=3|y|^{2/3}\). The functions \(p(x)=x^3\) and \(q(x)=0\) both satisfy it, since
At \(x=-1\), the initial values are \(p(-1)=-1\) and \(q(-1)=0\), so they are different. At \(x=0\), however, both functions equal \(0\): the curves meet. The right-hand side is not locally Lipschitz in \(y\) near \(0\); for example, the ratio \(\frac{|3|h|^{2/3}-0|}{|h-0|}=3|h|^{-1/3}\) is unbounded as \(h\) approaches \(0\). This example does not contradict the uniqueness result, because its required condition fails.
Using the Comparison Carefully
When comparing curves in a slope field, keep three questions separate: Where do they start? What slope does the equation assign at each starting point? And do the conditions that guarantee uniqueness apply? The first two questions describe local directions and positions. The third determines whether two paths can meet on the region under consideration.
Do not conclude that two curves cross just because one has a larger slope at a point. A slope comparison is local: it describes how the curves are directed there, not their entire paths. Likewise, do not conclude that two curves stay a fixed distance apart just because they have the same slope at one point. The separation can change later if the field assigns different slopes at their new locations.
Worked Example: Compare the Initial Gap and Directions
For \(y'=x-y\), consider a solution \(p\) starting at \((1,0)\) and a solution \(q\) starting at \((1,2)\). At \(x=1\), their slopes are
The first curve initially rises and the second initially falls. Their signed separation at the starting coordinate is \(q(1)-p(1)=2-0=2\). The derivative of their separation there is
So the initial gap is decreasing at \(x=1\), but this local calculation alone does not tell us its later size. Because \(F(x,y)=x-y\) is Lipschitz in \(y\), with \(|F(x,u)-F(x,v)|=|u-v|\), the uniqueness condition applies. The two solutions therefore cannot meet on a common interval where the condition holds.
Check: Substituting both starting points into \(x-y\) verifies the slopes \(1\) and \(-1\); subtracting those slopes verifies the separation derivative \(-2\).
Check Your Understanding
Answer each question using the differential equation and the comparison ideas above.
- For \(y'=2x-y\), what are the initial slopes for solutions starting at \((1,1)\) and \((1,4)\)?
- For \(y'=x\), can two solutions have different values but equal slopes at the same \(x\)? Explain briefly.
- If \(q(x)-p(x)>0\) and \((q-p)'(x)<0\) at one point, what does that say locally? What does it not establish?
- State the uniqueness condition that prevents two distinct solution curves from meeting on a common interval.
- In the equation \(y'=3|y|^{2/3}\), identify the different initial values and the later point where the displayed solutions meet.
