Differential equations are a special form of equation that includes a function and one or more of its derivatives.
They’re immensely helpful in many fields. Unfortunately, however, many students struggle with differential equations because they require the learner to use several different types of mathematical skills in order to solve them correctly, and if one or more of these foundational principles are missing, it can make the problem frustratingly difficult.
This short article will introduce you to the basic concepts necessary for learning differential equations, as well as provide you study tips on how to optimize your learning.
To begin, we'll discuss some simple principles that you should understand prior to trying to learn differential equations. These foundations are important, and include basic Calculus, Algebra, and Arithmetic.
Calculus is the branch of mathematics that deals with quantities and relationships between concepts like distance, velocity, acceleration, area, and more.
It's very useful when it comes to solving problems involving the rate at which quantities change. For example, if I know the speed and the acceleration of the vehicle, and I want to try and determine the distance, I would use simple calculus to solve for the answer.
Algebra is the branch of mathematics that deals with abstract operations on numbers and their properties.
Usually, letters are used as stand-ins for numbers as a form of generalization, allowing one algebraic equation to summarize the relationships between many different things.
The most common uses of algebra are in geometry, number theory, probability, statistics, and much more. Algebra is also used in many engineering fields such as mechanics and electronics.
Arithmetic is the branch of mathematics that deals with simple operations on numbers like addition, subtraction, multiplication and division.
It's used extensively in all sorts of calculations, and is usually one of the first skills you learn in school (so you'll probably have this covered).
Let's say that the function N(t) describes a population (N) with respect to time (t).
An example differential equation including this concept might be dN/dt = r*N(t).
In this equation, dN/dt stands for the rate of population change with respect to time, and we find that it equals the population N multiplied by some constant r.
In summary, the speed of population growth depends on the population itself multiplied by some other factor (like, for example, the growth rate!)
Why do we use differential equations?
Mathematicians and engineers often use differential equations any time they want to describe or model change.
For example, when population changes, as we saw above, the change is usually dependent on the current population.
More animals or people usually lead to more animals or people - this just intuitively makes sense. But by how much?
What's the difference in population growth rate when the population is 1,000 versus 100,000? Through the use of differential equations, we can accurately model this sort of change quantitatively and provide actual numbers describing the rates.
It turns out that any time something changes, you can use a differential equation to mathematically model that change!
This is very useful for problems in physics and other scientific fields that describe natural systems and their changes with time.
How do I learn differential equations?
The best way to learn is through practice.
Math requires an active learning mindset - just reading about differential equations isn't enough to learn how to actually solve them.
Below, we've provided a few basic problems for you to try and solve on your own. Underneath each problem is the solution.
We highly recommend that you try your best to solve the problem before looking at the solution, to give your mind a chance to 'flex' and grow.
1. Population growth
Human beings create more human beings through reproduction. Experiments have determined that the human growth rate, r, is 3.5.
Let's suppose that we're interested in determining the rate of change of the population, dP/dt, when the population is 1,000, and we can accurately model this through the differential equation dP/dt = r*P(t) (where P(t) is the population at time t).
What is the rate of population growth with respect to time when P(t) = 1,000?
dP/dt = 3.5 * 1000
dP/dt = 3500
2. Interest and financial growth rates
Suppose that you currently have $1,000,000 (M(t)=1,000,000).
You invest your money in a growth fund that provides an 8.00% interest rate per year (r=0.08). You are interested in determining the rate at which your money grows with respect to time, dM/dt.
Model an equation and solve it for dM/dt when M(t) = 1,000,000.
Since the rate at which money grows depends on both the amount of money you currently have, as well as some outside factor (interest rate), we can model the growth rate using a similar equation to what we saw in (1). This time, instead of population growth, we're looking at monetary growth.
Equation: dM/dt = r * M(t)
dM/dt = 0.08 * 1,000,000
dM/dt = 80000
Interpretation: because the time period we're using is yearly, dM/dt (the change in M with respect to the change in t) can be interpreted as 80,000 per year.
Note, however, that since most investment funds add the return to the principal year-over-year, your next year's growth will be higher (since it will now be acting on $1,080,000 instead of $1,000,000).
Tips for staying motivated while studying differential equations
Differential equations are hard to learn, and even harder to keep consistent on practicing. Below are some tips that will help you avoid fatigue and keep motivated.
1. Take frequent breaks.
Differential equations are a lot to take in, and your brain needs time to process all of this new information (as well as everything else you're learning at this stage in your life). Every couple of hours, take a 5-10 minute break. Walk around, get some water, move your body, and then return to the problem at hand.
2. Avoid burnout.
This is related to the first point, but is a little more broad. Differential equations, like anything else in life, should be enjoyed. Appreciate the beauty in the subject, even if you find it difficult.
3. Join a study group.
Studying with friends (or people with similar skill sets) can help keep you all on track and focused. It also gives you someone to reach out to for help when you get stuck.
4. Pace yourself.
There's no need to rush, and you will get overwhelmed if you try to do too much (or more likely, fail completely). Give yourself time to process the information, and don't be afraid to go back and review what you've learned.
5. Get plenty of sleep.
You may be able to get by for a while on little sleep, but as they say, "sleep is a basic survival requirement."
We hope the above article provided an effective introduction to differential equations. If you ever find yourself feeling stuck or in need of dedicated tutoring, engage a qualified math tutor from Tutor City.
We wish you the best on your mathematics journey & don’t hesitate to reach out if you have questions or comments.