IB Maths AI HLDifferential EquationsPaper 1 & 2~6 min read
Slope Fields
A slope field lets you see the solutions of a differential equation without solving it. At a grid of points you read off dydx and draw a short tangent line with that gradient. The pattern of little lines reveals the shape of every solution curve ā pick a point, follow the flow, and you have a sketch.
š What you need to know
What it is: a grid of short tangent lines, each with gradient dydx = g(x, y) at that point.
No solving needed: just substitute x and y into g(x, y).
Family of solutions: the field shows all of them; a condition picks one.
Tangents as flow lines: from a point, the solution curve flows along the local tangents.
Sketch, don’t trace: follow the directions ā don’t join the line segments or treat them as barriers.
Horizontal tangents: stationary points occur where dydx = 0.
Reading a slope field
š¤ Why can we draw it without solving?
The differential equation is a rule for the gradient: at any point (x, y) it tells you the slope of the solution passing through there. You don’t need the formula for the curve ā only its direction at each point. Plot enough of those directions and the curves announce themselves.
š§ “Substitute, draw the slope, follow the flow”
Substitute each grid point into g(x, y), draw a short line of that gradient, then to sketch a solution follow the flow of the tangents from your given point ā left and right.
Slope field for dy/dx = x ā y, with the solution through (0, 0)
Each segment’s gradient is x ā y at that point. The red curve flows along the tangents from (0, 0) ā a sketched solution, not a join-the-dots of the segments.
Sketching a solution & finding stationary points
š§ Recipe ā sketch a solution curve
Plot the given point on the slope field.
Follow the tangent at that point to set the initial direction.
Continue left and right, steering with the nearby tangents.
Keep it a sketch ā the shape, not an accurate plot.
Different start point ā a different solution curve.
Stationary points sit where the tangent is horizontal, i.e. dydx = g(x, y) = 0. Solving that equation tells you the line(s) or point(s) on which all stationary points lie ā even with no horizontal segments drawn.
ā Don’t trace the segments
The tangent lines are not guaranteed to lie on your solution curve.
You can cross tangent lines ā they aren’t boundaries.
Don’t simply join neighbouring segments end to end.
Worked examples
Examples 1ā2 use the differential equation dydx = ā0.4(y ā 2)1/3(x ā 1) eā(xā1)Ā²/25.
WE 1
Find the set of points where the solutions have horizontal tangents.
Horizontal tangents mean dydx = 0. A product is zero when a factor is zero ā and the exponential is never zero.
eā(xā1)Ā²/25 ā 0, so ignore it(y ā 2)1/3 = 0 ā y = 2(x ā 1) = 0 ā x = 1horizontal tangents wherever y = 2 or x = 1
WE 2
Describe how you would sketch the solution through the point (0, ā8) on the given slope field.
Use the point and the local tangents as flow lines.
plot (0, ā8)follow the tangent direction there, extending left and rightsteer with neighbouring segments; cross them if the flow doesa single curve through (0, ā8) ā sketched, not traced
WE 3
For dydx = sin(x ā y), find the lines on which all stationary points lie.
Set the gradient to zero and solve.
sin(x ā y) = 0 ā x ā y = 0, Ā±Ļ, Ā±2Ļ, ā¦stationary points lie on y = x Ā± kĻ (k = 0, 1, 2, ā¦)
WE 4
Find the gradient of the slope-field segment at (2, ā1) for dydx = x ā y.
Substitute the coordinates into g(x, y).
g(2, ā1) = 2 ā (ā1)gradient = 3
WE 5
For dydx = xy, where are the horizontal tangents?
A product equals zero when a factor is zero.
xy = 0 ā x = 0 or y = 0horizontal tangents along the x-axis and the y-axis
š” Top tips
To get a gradient, just substitute the point into g(x, y).
Solve dydx = 0 for stationary points, even if no horizontal segments are drawn ā there are marks for the correct line(s).
An exponential factor is never zero ā discard it when finding where the gradient is zero.
Follow the flow from the given point in both directions.
Different points give different curves ā the shape depends on the start.
Sketch only ā accuracy isn’t expected.
ā Common mistakes
Joining the segments instead of sketching a smooth flow line.
Treating tangents as walls ā solution curves can cross them.
Forgetting to solve dydx = 0 when no horizontal segments appear.
Setting an exponential to zero ā it never is.
Drawing one curve for all points ā each start point gives its own.
Next up ā Euler’s Method. A slope field shows the direction of solutions everywhere; Euler’s method turns that idea into numbers. Starting from the given point, it takes small steps along the local gradient ā exactly the tangent directions you’ve been reading off ā to build an approximate numerical solution, step by step.
Need help with Differential Equations?
Get 1-on-1 help from an IB examiner who knows exactly what Paper 1 & 2 are looking for.
