Qualitative and numerical approaches
When formulas fail, pictures and computers still reveal how solutions behave.
Slope fields
For \(y' = f(x, y)\), draw tiny segments with slope \(f(x, y)\) at grid points. Solution curves are tangent to the segments everywhere. Even without solving, you can see where solutions rise, fall, converge, or diverge. Software draws slope fields instantly.
Isoclines
An isocline is a curve along which the slope \(f(x, y)\) is constant. For \(y' = x - y\), the isoclines \(x - y = c\) are lines; along \(y = x - 1\), every slope is 1, and indeed \(y = x - 1\) is itself a solution.
Equilibria and stability
For autonomous equations, equilibria are constant solutions. If \(f'(y^*) < 0\), the equilibrium \(y^*\) is stable (nearby solutions approach); if \(f'(y^*) > 0\), it is unstable. This derivative test is the one-dimensional version of linear stability analysis used for systems later.
Bifurcations
When a parameter changes, equilibria can appear, disappear, or swap stability, an event called a bifurcation. For a harvested population \(P' = rP(1 - P/K) - H\), increasing the harvest \(H\) past \(\frac{rK}{4}\) makes both equilibria vanish and the population collapses.
Euler's method
Approximate the solution by stepping along tangent lines: \(y_{n+1} = y_n + hf(x_n, y_n)\). For \(y' = y\), \(y(0) = 1\), step \(h = 0.1\) gives \(y(1) \approx 1.1^{10} \approx 2.594\), compared with \(e \approx 2.718\). The error shrinks roughly in proportion to \(h\): it is a first-order method.
y_{n+1} = y_n + h\,f(x_n, y_n)Improved methods
Heun's method averages the slope at the start and an estimated end of each step, giving second-order accuracy. The classical fourth-order Runge-Kutta method (RK4) combines four slope estimates per step and is accurate enough for most practical problems with modest step sizes.
Stiffness
Some equations combine very fast and very slow behaviour, like chemical reactions on different timescales. Explicit methods then need tiny steps to stay stable. Implicit methods, which solve an equation at each step, handle these stiff problems efficiently.
Common mistakes
Taking steps too large, so numerical solutions oscillate or blow up. Trusting a numerical answer without checking how it changes when the step size is halved. Reading stability from a slope field incorrectly near equilibria.
Key takeaways
Slope fields and isoclines show solution behaviour graphically. The sign of \(f'(y^*)\) decides stability of equilibria, and bifurcations occur as parameters change. Euler's method is simple but first-order; Heun and RK4 are more accurate; stiff problems need implicit methods.
Flashcard
Segments with slope f(x, y)
Check your understanding
Using Euler's method with h = 0.5 on y' = 2y, y(0) = 1, what is the estimate of y(0.5)?