Differential equation (math)

MAT 275 Online Activity 3

A differential equation is called autonomous when it does not explicitly depend on the

independent variable. For example, 𝑦′ = 𝑦 is autonomous but 𝑦′ = 𝑦 + 𝑡 is not.

If the independent variable is time, then we can explain what autonomous really means in

terms of application: an autonomous equation does not know what time it is.

Take for example the equation 𝑦′ = 𝑘𝑦 with a positive constant k. It describes exponential

growth, say, of a bacteria culture. The equation says that the growth rate of the culture is

proportional to how many bacteria there are already. The more bacteria there are, the faster

the culture grows.

It would not make any sense to have an explicit t dependence in that equation, because

bacteria don’t know what time it is. A 19th century and a 21st century scientist cultivating the

same bacteria under the same condition would observe the exact same growth curves relative

to their own starting times of the experiment.

Differential equations that express natural laws with time being the independent variable are

always autonomous, because to the best of our current knowledge, the laws of nature do not

change over time. The same experiment, conducted yesterday, today or tomorrow, must

always produce the same outcome relative to the starting time.

This leads us directly to the following fact: if 𝑦(𝑡) is a solution of an autonomous differential

equation, then any time - shifted version of that function, i.e. 𝑦(𝑡 − 𝑐), must also be a solution.

It must be, because the equation does not know what time it is. If it allows for a behavior at one

time, it must allow for the same behavior at all times.

1. The first goal of this activity is to demonstrate that this hypothesis is true for first order

equations. Let us start by assuming that the given first order autonomous equation is

𝑦′ = 𝑓(𝑦).

Demonstrate, using the rules of differentiation, that if 𝑦1(𝑡) is a solution of that

equation, then 𝑦2(𝑡) = 𝑦1(𝑡 − 𝑐), where c is a constant, is a solution as well. You must

show all algebraic steps.

2. The fact that shifted solutions of autonomous equations are again solutions has an

important consequence for the general solution of such equations: the general solution

of an autonomous equation is shift-invariant. If you introduce a time shift into the

general solution it’s still the general solution. For example, we know that the general

solution of 𝑦′ = 𝑦 is 𝑦 = 𝐶𝑒𝑡 . By the theorem we just learned, then, 𝑦 = 𝐶𝑒𝑡−1

is also the general solution.

Use algebra to demonstrate this. Show that every function of the form 𝐶𝑒𝑡−1 is also a

function of the form 𝐶𝑒𝑡 and vice versa. Do not just write equations. You must explain

your reasoning.

3. Shift-invariance of the general solution can lead to dramatic computational

simplifications when you have to satisfy initial conditions. Let us consider the equation

𝑦′′ + 2𝑦′ + 10𝑦 = 0.

Its general solution is 𝑦(𝑡) = 𝑒−𝑡 (𝐴 cos 3𝑡 + 𝐵 sin 3𝑡 ).

Say we want to find the particular solution that satisfies the conditions 𝑦(1) = 0 and

𝑦′(1) = 1. Direct substitution into the general solution leads to a system of equations

for A and B that is very complicated.

Demonstrate how, by exploiting shift-invariance of the general solution, you can find the

desired particular solution very easily.