From a Differential Equation to a Local Slope
A differential equation such as \(y'=f(x,y)\) specifies a rule for the derivative of a solution \(y\). Once a point \((x,y)\) is given, evaluating \(f\) at that point gives the slope that a solution curve passing through it must have there.
This tutorial focuses on reading those slopes at specified points and organizing them in a table. The result describes the curve’s direction locally at each point; it does not by itself provide a formula for the entire solution curve.
The coordinates must be substituted in the correct order: \(x_0\) is the input for \(x\), and \(y_0\) is the input for \(y\). The evaluated number is a slope, not the value of the solution \(y\) at that point.
Evaluating the Right-Hand Side
To find the slope at a specified point, substitute both coordinates into the right-hand side of the equation, then simplify. If the formula contains a denominator or another restricted expression, first check that it is defined at the point.
Worked Example: Find Slopes at Three Points
For \(y'=2x-y\), find the slopes at \((1,2)\), \((-1,1)\), and \((2,-1)\).
At \((1,2)\), substitute \(x=1\) and \(y=2\):
The slope at \((1,2)\) is \(0\).
At \((-1,1)\), substitute \(x=-1\) and \(y=1\):
The slope at \((-1,1)\) is \(-3\).
At \((2,-1)\), substitute \(x=2\) and \(y=-1\):
The slope at \((2,-1)\) is \(5\).
Check: Substitution into \(2x-y\) gives \(0\), \(-3\), and \(5\), respectively, at the three stated ordered pairs. Each computed value is the derivative required by the equation at that point.
Each result can be described geometrically. A slope of \(0\) means the tangent line is horizontal. A negative slope means the tangent line falls as \(x\) increases; a positive slope means it rises. These statements describe the tangent direction at the specified point, not necessarily the overall behavior of a solution over a longer interval.
Recording Slopes in a Table
When several points are supplied, a table keeps the coordinates and their corresponding slopes together. A useful layout includes one column for the point and one for the value of \(f(x,y)\). The point should be written as an ordered pair so the substitution order remains clear.
Worked Example: Build and Read a Slope Table
For \(y'=(x+1)(y-2)\), evaluate the slope at \((0,0)\), \((0,3)\), \((2,0)\), and \((2,3)\). Then describe the tangent direction at each point.
At \((0,0)\):
At \((0,3)\):
At \((2,0)\):
At \((2,3)\):
| Point \((x,y)\) | Slope \(f(x,y)\) | Local tangent direction |
|---|---|---|
| \((0,0)\) | \(-2\) | Falls as \(x\) increases |
| \((0,3)\) | \(1\) | Rises as \(x\) increases |
| \((2,0)\) | \(-6\) | Falls more steeply than at \((0,0)\) |
| \((2,3)\) | \(3\) | Rises more steeply than at \((0,3)\) |
Check: In each row, multiplying the substituted factors reproduces the recorded slope: \(1(-2)=-2\), \(1(1)=1\), \(3(-2)=-6\), and \(3(1)=3\). The descriptions match the signs and relative sizes of those values.
The table reports a slope at each listed point only. It does not assert that a solution passes through every listed point, nor does it connect the values into a single curve. If a solution does pass through one of the points, its tangent there must have the slope shown in that row.
Why the Evaluated Value Is the Local Direction
The interpretation follows directly from what it means for a function to solve a differential equation. A differentiable function \(y(x)\) is a solution on an interval when, at every \(x\) in that interval, its derivative equals the right-hand side evaluated using the function’s current value: \(y'(x)=f(x,y(x))\). At \(x=x_0\), if \(y(x_0)=y_0\), substitution gives the derivative at that point.
The derivative is the slope of the tangent line. It also gives the first-order local approximation near \(x_0\). For a small change \(h\), differentiability means that the change in \(y\) is approximately the derivative at \(x_0\) multiplied by \(h\):
This approximation explains why a slope value describes the nearby direction of a solution curve. For a small positive \(h\), a positive slope predicts a value above \(y_0\), while a negative slope predicts a value below \(y_0\), to first order. The approximation is local: it does not claim that the curve is a straight line or that the same slope persists farther away.
Including Units in a Rate Model
In an applied model, the slope can carry units. For example, if \(t\) is measured in years and \(P\) in thousands of people, then \(dP/dt\) is measured in thousands of people per year. The numerical value from substitution is therefore a rate in those units.
Worked Example: Interpret Slopes with Units
Suppose \(P'=0.2t+0.5P-4\), where \(t\) is measured in years and \(P\) in thousands of people. Evaluate the slope at \((t,P)=(2,4)\) and \((5,6)\), and interpret the results.
At \((2,4)\), substitute \(t=2\) and \(P=4\):
The rate there is \(-1.6\) thousand people per year. At that point, a solution passing through it has a tangent that falls as time increases.
At \((5,6)\), substitute \(t=5\) and \(P=6\):
The rate there is \(0\) thousand people per year, so the tangent to a solution at that point is horizontal.
Check: The first substitution gives \(0.4+2-4=-1.6\); the second gives \(1+3-4=0\). Both results retain the rate units of thousands of people per year because the equation defines \(P'\).
A Common Substitution Mistake
A frequent error is to use only the \(x\)-coordinate, or to reverse the coordinates, when evaluating \(f(x,y)\). The equation \(y'=2x-y\) depends on both variables. At \((1,2)\), the correct substitution is \(2(1)-2=0\). Reversing the coordinates gives \(2(2)-1=3\), which is not the slope at \((1,2)\); it evaluates the expression at \((2,1)\).
Another distinction matters: \(y'\) is the slope, while \(y\) is the vertical coordinate. At \((1,2)\) in this example, the solution’s value is \(y=2\), but its derivative is \(y'=0\). Reporting \(2\) as the slope confuses the coordinate with the rate of change.
Using a Slope Table Carefully
A table of evaluated slopes is a compact way to compare the local tangent directions specified by a differential equation. For each row, it is useful to ask three questions:
- What ordered pair is being evaluated?
- What exact value does the right-hand side give there?
- What does that value say about the tangent at that point?
A numerical slope also gives information about steepness: for example, a slope of \(-6\) has a larger magnitude than a slope of \(-2\), so its tangent is steeper. But a table value alone describes only its own point. It should not be extended to nearby points unless the equation is evaluated there as well.
The essential process is straightforward: identify the point, substitute both coordinates into the right-hand side, simplify, and interpret the result as a tangent slope. This turns the symbolic rule \(y'=f(x,y)\) into precise local direction information without confusing a slope with an entire solution curve.
Check Your Understanding
For each question, show the substitution before stating the slope.
- For \(y'=x-2y\), what is the slope at \((3,1)\)?
- For \(y'=x+y\), find the slopes at \((0,2)\) and \((2,0)\). Record them in a two-row table.
- For \(y'=(x-1)y\), what is the slope at \((1,4)\), and what tangent direction does that value indicate?
- For \(y'=y/(x+2)\), is the slope defined at \((-2,3)\)? Explain using the expression’s denominator.
- If a solution passes through \((x_0,y_0)\) and \(f(x_0,y_0)=-3\), what does that tell you about its tangent there as \(x\) increases?
