Examples from Numerical solutions

A family of translated curves

For

every point with the same $x$-coordinate has the same slope. Integrating gives

The curves through $y(0)=1$ and $y(0)=3$ are vertical translates of one another. The direction field displays this common slope pattern even though the initial condition chooses only one curve.

A visibly large step

For

the first point is $(x_0,y_0)=(0,1)$. Since $f(0,1)=0$, the next point is $(1,1)$. The following slopes are $f(1,1)=1$ and $f(2,2)=4$, giving

The exact solution is $y=1+x^3/3$, whose value at $x=3$ is $10$. The step size is too large to track the curvature accurately.

A smaller step

For

Euler's update gives

The exact solution is

whose value at $x=0.3$ is approximately $1.3997$. The Euler value $1.362$ is below it by about $0.0377$.

Comparing Euler and Heun

Consider

whose exact solution is

With $h=0.2$, $f(0,0)=1$. Euler predicts $u_1=0.2$; the predicted endpoint slope is $f(0.2,0.2)=1/1.04-2(0.2)^2\approx0.88154$. Heun therefore gives $y_1\approx0.2(1+0.88154)/2=0.18815$, closer to the exact value $y(0.2)=0.2/1.04\approx0.19231$ than Euler's $0.2$. Repeating the updates on $[0,1]$ gives these maximum grid-point errors:

Reducing $h$ lowers the error for both methods and Heun keeps the smaller error.