A Love-Hate Relationship
UBC Calculus Online Course Notes

A system of linked differential equations

Patterns of derivatives in the Trigonometric Functions

In our investigation of the trigonometric functions so far, we saw that the derivatives of the trigonometric functions satisfy an interesting relationship:

\[ \frac{d\cos(t)}{dt} = - sin(t) \] \[ \frac{d \sin(t)}{dt} = \cos(t) \]
Thus, if we name these functions $ y(t)=\sin(t), x(t)=\cos(t) $ , we find that they satisfy the following connected relationships:

\[ \frac{dx}{dt} = -y \] \[ \frac{dy}{dt} = x \]

This type of relationship, in which two functions $ y(t), x(t) $ are related to their own derivatives is another example of a set of differential equations. We see that the two equations are connected to one another: if we know $ x(t) $ , we know how $ y(t) $ changes, and vice versa. This type of linked set of differential equations is called a system of coupled differential equations. The above set is one of the simplest examples of such a system. (In future courses on differential equations you would learn that this system is said to be linear because changes are simply proportional to the values of the functions. We will not delve into this terminology here.)

Another relationship that we can easily see is this: If we take second derivatives, we find that

\[ \frac{d^2\cos(t)}{dt^2} = - \frac{d sin(t)}{dt} = - cos(t) \] \[ \frac{d^2 \sin(t)}{dt^2} = \frac{d \cos(t)}{dt} = - sin(t) \]

Thus, each one of the functions also satisfies the relationship

\[ \frac{d^2y}{dt^2} = - y \]

(Note: you might wonder why we don't add the equation

\[ \frac{d^2x}{dt^2} = - x \]

but please realize that this is exactly the same relationship: It states that the second derivative of the given function is proportional to the function itself (with -1 being the constant of proportionality).

The equation above is also a differential equation. Since it involves a second derivative, it is usually called a second order differential equation . We have just made an interesting connection: the trigonometric functions $ \sin(t), \cos(t) $ are solutions to a pair of simple (first order) differential equations. They are also solutions to a a second order differential equation. (It turns out that these two things are equivalent.) We will later see that this has great implications to phenomena from many important branches of science.


For your consideration:

  1. (1) Show that not just the simple trig functions, but also combinations of these functions satisfy the same relationships. Take the derivatives of the functions

    \[ x(t)=A \sin(t) + B \cos(t) \] \[ y(t)=B \sin(t) - A \cos(t) \]
    for A, B constants and show that they satisfy the system of equations, as well as the second order differential equation.

  2. (2) What do you suppose might be needed to determine values of these constants in a given situation?
We will study one particular (slightly whimsical) example in which this kind of system of differential equations occurs on the next page.