Approximating differential equation with Euler’s method

It looks like it’s time for differential equations.

So, let’s say that you have a differential equation like the one below and you are striving trying to find an analytical solution. Chill out, maybe take a walk outside or ride your bike and calm down: sometimes an analytical solution does not exist. Some differential equations simply don’t have an analytical solution and might drive you mad trying to find it.

Fortunately, there is a method to find the answer you may need, or rather, an approximation of it. This approximation sometimes can be enough of a good answer.

While there are different methods to approximate a solution, I find Euler’s method quite easy to use and implement in Python. Furthermore, this method has just been explained by Sal from Khan Academy in a video on YouTube which is a great explanation in my opinion. You can find it here.

Here below is the differential equation we will be using:

and the solution (in this case it exists)

Here is the Python code algorithm. Note that the smaller the step (h), the better the approximation (although it may take longer).

	# Euler's method for approximating differential equations
	import math
	import matplotlib.pyplot as plt
	# Your differential equation. It should be in the form: dy/dx= f(t,y)
	# Remember that f returns the value of the slope in that point
	def f(t,y):
	return 2-math.e**(-4*t)-2*y
	def solution(t):
	return 1+0.5*math.e**(-4*t)-0.5*math.e**(-2*t)
	t0 = 0
	y0 = 1
	# Step size, the smaller the better, however it may take longer to compute
	h = 0.005
	x = [t0]
	y = [y0]
	y2 = [y0]
	errors = []
	while t0 < 5:
	m = f(t0,y0)
	y1 = y0 + h*m
	t1 = t0+h
	x.append(t1)
	y.append(y1)
	solut = solution(t1)
	y2.append(solut)
	error = abs(y1-solut)/y1*100
	errors.append(error)
	print("Actual t","predicted y","Percentage error %",sep=" ")
	print(t1,y1,error)
	t0=t1
	y0=y1
	print("Mean of error:",sum(errors)/len(errors),"%",sep=" ")
	plt.plot(x,y2,label="Function")
	plt.plot(x,y,label="Approximation")
	plt.legend(loc='upper right')
	plt.title("Euler's method approximation, h="+str(h))
	plt.show()

view raw diff_eq_Euler_approx.py hosted with ❤ by GitHub

Here are some approximations using different values of h:

Hope this was interesting.