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Slope Fields and Euler's Method
Calculus 2 · Chapter 5: Differential EquationsSection 5.2 · lesson 23 of 5410 min · free
← Differential Equations: Order, Solutions, and IVPsNext lesson: Separable Differential Equations →
In this lesson
Seeing and approximating solutions you cannot write down.
- Read a differential equation as a slope at every point
- Sketch solution curves from a slope field
- Step along tangent lines with Euler's method
- Trade step size against accumulated error
Transcript
Everything said in the video, section by section (1,493 words). Read along, or search it.
MAT-265 - Calculus 2 · Section 5.2
Slope Fields and Euler's Method
A biologist modeling a population writes down the differential equation that governs its growth, expecting to solve it and predict the future. But many differential equations — even simple-looking ones, like d y d x equals x minus y squared — have no solution expressible in elementary functions. There is no formula to find. No clean curve to plot. And yet the equation still contains complete information about how the system behaves. The challenge is extracting that information without an explicit solution. Two tools make it possible. A slope field reveals the qualitative shape of every solution at a glance — turning the equation itself into a picture. And Euler's method provides quantitative estimates, marching step by step along a solution using nothing but the differential equation and a starting point.
The Problem
When a Solution Cannot Be Written Down
Everything in this section rests on one observation. A first-order differential equation in the form d y d x equals f of x comma y has a special interpretation. It gives the slope of the solution curve at every point x comma y in the plane. Think about what that means. You don't need to solve anything. Pick any point. Plug its coordinates into f. And you immediately know how steeply the solution through that point is rising or falling. That single observation is the foundation for everything that follows.
Slope Fields
Seeing Every Solution at Once
Since the equation specifies the slope at any point, you can visualize it by drawing a short line segment of that slope at many points across a grid. The resulting collection of segments is the slope field. No solving is required. Each segment is computed by plugging the point's coordinates into f. Take the equation d y d x equals x minus y. At the point two comma zero, the slope is two minus zero. Two. Steep and rising. At zero comma two, the slope is zero minus two. Minus two. Steep and falling. And along the whole line y equals x, the slope is zero — so those segments are flat. Plot hundreds of these, and you get a flowing pattern that the solution curves must follow. Like iron filings aligning with a magnetic field. Now, how do you read one? A solution curve is one that stays tangent to the segments everywhere it passes. So to sketch a particular solution, start at the initial-condition point, and draw a smooth curve that flows with the segments — always matching their direction. Different starting points trace different members of the solution family. And crucially, solution curves never cross. Because a single point has only one slope, so only one curve can pass through it.
What the Picture Tells You
Qualitative Facts, No Formula Needed
Slope fields reveal qualitative features that no formula is needed to see. First. Equilibrium solutions. Horizontal lines where f of x comma y is zero everywhere along them — solutions that stay constant forever. Second. Long-term behavior. Whether solutions rise, fall, level off, or blow up as x grows. In the figure, notice how all the solutions are drawn toward the line y equals x minus one. That's visible immediately, without solving anything. Third. Stability. Whether nearby solutions converge toward a curve, or diverge away from it. Now here's something genuinely worth thinking about. A slope field is built entirely from the differential equation, with no solving involved. Yet it shows the behavior of every solution simultaneously. How can a picture contain more information than any single formula? Because the field encodes the rule that all solutions obey — rather than one particular solution. Selecting an initial condition is like dropping a leaf onto a flowing stream. The leaf's path is determined by two things. The current — that's the field. And where it lands — that's the initial point. Change the landing spot, and the same field carries it along a different solution.
Euler's Method
Stepping Along Tangent Lines
A slope field shows the shape of solutions. But it doesn't give you numerical values. Euler's method fills that gap, with an idea so simple it's almost obvious. If you know the slope at the current point, take a small step in that direction. Then recompute the slope, and repeat. Each step follows the tangent line for a short horizontal distance, before correcting course. Begin at the initial point x naught comma y naught. Choose a step size h — the horizontal distance per step. The slope there is f of x naught comma y naught. So moving h to the right and following that slope lands you at a new y. In words: new y equals old y, plus slope times run. Repeat from the new point, and the general stepping formula emerges. x n plus one is x n plus h. And y n plus one is y n, plus h times f of x n comma y n. Let's do one. Use Euler's method with step size zero point five to approximate y at two, for d y d x equals x plus y, given y at zero equals one. Four steps. Step one. The slope is zero plus one, which is one. So x moves to zero point five, and y becomes one plus zero point five, which is one point five. Step two. The slope is now zero point five plus one point five. Two. So y becomes two point five. Step three. Slope is one plus two point five. Three point five. y becomes four point two five. Step four. Slope is one point five plus four point two five. Five point seven five. And y becomes seven point one two five. So Euler's method estimates y at two is about seven point one two five. And notice the work organizes naturally into a table. Column by column, row by row.
Accuracy
Step Size and the Accumulation of Error
Now — how good is that estimate? For comparison, the exact solution of this equation is y equals two e to the x, minus x, minus one. Which gives y at two equals two e squared minus three. About eleven point seven eight. Our Euler estimate of seven point one two five is well short. So what went wrong? Euler's method is only an approximation, because each step replaces a curving solution with a straight tangent line. Over one step, the error is small. But errors accumulate. Each step starts from a point that is already slightly off — so inaccuracies compound as the stepping proceeds. Look at the picture. Because the true solution curves away from each tangent line, the polygonal path falls progressively below it. And the gap — the accumulated error — widens with every single step. The size of that error is controlled by the step size h. A smaller h means shorter tangent steps that hug the curve more closely, reducing error. But it also requires more steps, and more arithmetic, to cover the same interval. That's the central trade-off of the method. Accuracy versus effort. And the relationship is roughly linear. Halving the step size halves the accumulated error. For this reason Euler's method is called a first-order method. Redo that example with h equal to zero point two five — eight steps instead of four — and you get about eight point nine two. Noticeably closer to the true eleven point seven eight.
Key Takeaways
What to Carry Forward
One more, to show where this really earns its keep. Take two steps of Euler's method with h equal to zero point one, for d y d x equals y squared minus x, with y at zero equal to one. Step one. The slope is one squared minus zero. One. So x becomes zero point one, and y becomes one point one. Step two. The slope is one point one squared, minus zero point one. That's one point two one minus zero point one. One point one one. So x becomes zero point two, and y becomes one point one plus zero point one one one. One point two one one. The estimate is y at zero point two is about one point two one one. And here's the point. This equation has no elementary solution. So Euler's method — or a refinement of it — is the only practical way to obtain a value at all. Five takeaways. A first-order equation assigns a slope to every point. Plotting short segments of those slopes gives the slope field — a picture of all solutions at once. Sketch a solution by starting at the initial point and drawing a curve that stays tangent to the segments. And solution curves never cross. The field reveals equilibria, long-term behavior, and stability, without solving anything. Euler's method steps along tangent lines. y n plus one equals y n, plus h times f of x n comma y n — using only the equation and a starting point. And error accumulates, because each step is a straight approximation to a curve. Halving h roughly halves the error, at the cost of twice the work. Next up is section five point three — Separable Equations. The first family we can actually solve exactly.
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