The simplest differential equation is $\frac{df}{dx} = f(x)$.

In other words, the derivative of the function is proportional to itself .

Show that $f(x)$ must equal the infinite sum
$$
f(x) = 1+x+\frac{1}{2}x^2 +\frac{1}{6}x^3 +\dots
$$

f(x) = 1+x+ 1
2
 x2 + 1
6
 x3 +...

Writing functions as a sum of simple powers of x [called a power series] is a very powerful technique.

Now, suppose that you wish to do some trigonometry and the sin and cos buttons are broken on your calculator. Use the fact the sin(x) and cos(x) both solve the simple harmonic motion equation


d2f
dx2
 +f(x) = 0

to find the power series for these two functions up to the 6th power of x (here x is measured in radians). [extension: can you spot the form of the infinite power series?]

You can now estimate trigonometrical values without using the sin or cos button. Test the accuracy of your series for values of x between 0 and pi/2.
Discussion points: Do you think that your calculator stores values of sin and cos, or works them out on demand. Would your series provide an efficient way of evaluating the numerical values of sin(x) and cos(x)?