There are many examples of differential equations that cannot be solved analytically – in fact, it is very rare for a differential equation to have an explicit solution. Euler’s Method is a way of numerically solving differential equations that are difficult or that can’t be solved analytically. Euler’s method, however, still has its limitations.

For a differential equation $y^{\prime}=f(x,y(x))$ with initial condition $y(x_{0})=y_{0}$ we can choose a step-length $h$ and approximate the solution to the differential equation by defining $x_{n}=x_{0}+nh$ and then for each $x_{n}$ finding a corresponding $y_{n}$ where $y_{n}=x_{n-1}+hf(x_{n-1},y_{n-1})$. This method works quite well in many cases and gives good approxiamtions to the actual solution to a differential equation, but there are some differential equations that are very sensitive to the choice of step-length $h$ as the following demonstrates.

Let’s look at the differential equation $y^{\prime}+110y=100$ with initial condition $y(0)=2$.

This differential equation has an exact solution given by $y=1+\mathrm{e}^{-100t}$ but this example is a very good example which demonstrates that Euler’s method cannot be used blindly. Let’s look at what happens for a few different step-lengths.

For the step-length $h=0.019$ step-length we get the following behaviour

Behaviour of numerical solution of an ODE

The red curve is the actual solution and the blue curve represents the behaviour of the numerical solution given by the Euler method – it is clear that the numerical solution converges to the actual solution so we should be very happy. However, look what happens when the step-length $h=0.021$ is chosen

Behaviour of numerical solution of an ODE

Again the actual solution is represented by the red line which on this diagram looks like a flat line because the blue curve gets bigger and bigger as you move along the $x$-axis. So a change of just $0.002$ in the step-length has completely changed the behaviour of the numerical solution. For a step-length $h=0.03$ the graph would look as follows

Behaviour of numerical solution of an ODEThe actual solution can barely be seen and the numerical solution gets out of control very quickly – this solution is completely useless – the scales on the $y$-axis are enormous and increasing the step-length only makes this worse. What has happened?

It can be shown by induction that for $n \in \mathbb{N}$ that $y_{n}=1+(1-100h)^{n}$. This converges only for $h<0.02$ and diverges for $h>0.02$. $h=0.02$ is a limiting case and gives an oscillating numerical solution that looks as follows

Behaviour of numerical solution of an ODE

For this particular example for $h<0.02$ and as the step-length gets closer to $0$ the solution will converge faster and for $h>0.02$ as the step-length increases the solution will diverge more rapidly.

So even though we have Euler’s method at our disposal for differential equations this example shows that care must be taken when dealing with numerical solutions because they may not always behave as you want them to. This differential equation is an example of a stiff equation – in other words, one that is very sensitive to the choice of step length. In general as the step-length increases the accuracy of the solution decreases but not all differential equations will be as sensitive to the step-length as this differential equation – but they do exist.

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