Differentiating results in us knowing how large y-output changes compared to qan amount of x-input in the function f(x)=y.
Integrating is, uh, just “the summation” of all y-outputs as a result of x-inputs.
Now, what does a DE tell me? And what does solving it do?
In: Mathematics
Sometimes knowing how fast something is changing is of interest. Here is one example.
You can have an equation for position, where an object is in space.
If you take the derivative of that equation, you get an equation of how fast that object is moving at any given moment in time.
If you take the the derative of that, you have an equation of how that object is accelerating.
A differential equation is an equation with at least one derivative in it.
A derivative deals with rates of change. So, a differential equation says that a rate of change is equal to something.
In early math courses, solutions are often a single number. Later on, solutions can be functions or equations.
For differential equations, we often want to solve for what function has the derivative that has the properties described by that differential equation.
For example, you may observe in nature that something is happening at a certain speed. You can write that down as a differential equation, because speed is the derivative of position (speed is how fast the position changes). Solving this differential equation means finding the equation that describes the position.
It is just like an algebraic equation. But your unknown value also depends how fast it is changing (or vise versa). For example; you put x amount of rabbits in a confined space. They keep breeding. You wanto to know the number of rabbits in week, say, 10. The number of rabbits that exist on any day depends on how fast the number of rabbits change. While the amount of rabbits increase, the rate of increase also changes (more rabbits means more breeding).
An ELI5 way of talking about it would be saying sometimes you know how fast a thing is ***changing*** but you really want to know the thing itself.
For example, position. Maybe you don’t know your actual position, but you do have something that measures your speed (change in position). You can solve the equation involving the change in position to determine your actual position, as long as you know at least one value of the position at a given time.
Another example might be heating up water or something. You don’t have a thermometer, but if you know the starting temperature (room temperature) and how much heat you’ve put in, you can still solve for the temperature.
All math, at its core, is about predicting behavior. If I know [thing 1], then I know [thing 2] because of [behavior].
Differential equations deal with the relationship of [rate of change] of [behavior]. A lot of the times this is dependent on time, how fast is something is running out money, etc.
It doesn’t have to be time, it can be rate of change of x with respect to y. One example is how quickly elevation of a hill changes
Let’s imagine that you have a bathtube with a leak in it. You want to take a bath without all of the water running out, so you are going to run the faucet to add more water into the tube to offset the leak. However, the amount of water in the tub determines how faster water is going to leak out through the leak. This kind of problem is know as a “related rates” problem and is a fairly simple example of a differential equation. What we want to solve for is the rate of water we need to add, but that is dependant the rate of change of the leak, which is depend on the current water level. So:
y’ = a*x’ + b*x
Where a and b are some constants. After solving the DE, we will get an equation for y’ that is dependant on just x.
This is a fairly simple example, but what DE’s tend to be really useful is solving motion control problems. The one variable you have control over is the acceleration applied to the motion and you want to keep the system as a specific velocity or distance. Vehicle control is the obvious application, but the same math works for pretty much any physical system. The circuitry keeping your oven at a constant temperature is doing the same kind of operations.
It’s been a loooong time since I did differential equations and I’m sure I won’t be able to solve anything if you ask me now (without reviewing) but here’s my 5 cents.
You already knkw about derivatives. As other comments have already said, a differential equation is an equation that has one or more derivatives in it. You already know about integrals – integrals are actually the way to solving differential equations. The simplest integral you can find ( integral(1″dx) = x + C ) is actually the solution for the differential equation dy/dx = 1.
So in the most basic of ways, what a differential equation at its simplest is really just the same as a derivative. It shows you the rate of change of something with respect to another thing.
Now how about those with 2nd derivatives and more complex stuff you see when you start at the differential equations course your engineering school offers you? Turns out the world is not so simple that you can solve for all those things using just your simple derivatives and integrals. For more complicated events you need to use more complicated equations and more complicated solutions. The number of derivatives in each equation may change and there might be more clutter in them, but in essence what they wanna show you is the same – how something changes with respect to another thing. And solving those equations will tell you how much of that one thing is there when there’s this much of that another thing.
Other comments have already described a lot so maybe my contribution can be the introductory example (assuming knowledge of calculus): f(x) = f'(x). This is a DE, and it can be solved by dividing both sides by f(x) to get f'(x)/f(x) = 1 and then integrating both sides with respect to x. On the right this just becomes x + C_1, and on the left we can use u-substitution by letting u = f(x) so that du = f'(x)dx, meaning the integral becomes the integral of 1/u with respect to u. This is ln(u) + C_2 (or ln(|x|) + C_2 depending on how rigorous we’re being). So we can substitute f(x) back in to get ln(f(x)) + C_2 = x + C_1. Then solve for f(x):
ln(f(x)) + C_2 = x + C_1
exp(ln(f(x)) + C_2) = exp(x + C_1)
exp(ln(f(x)))exp(C_2) = exp(x)exp(C_1)
exp(ln(f(x))) = exp(C_1)/exp(C_2) * exp(x)
f(x) = Ce^x (where we usually just write C = e^C_1 / e^C_2 )
And there we have it, we have solved the differential equation where a function’s derivative equals the original function, to find out what functions satisfy such an equation.
A differential equation is a relation between one or more derivatives of a function. A key thing to pin down is that the solution to a diff equation is a function.
The simplest one I can imagine is f'(x) = f(x) and the solution is e^(x) since d/dx e^(x) = e^(x) (this is technically not the defining property of exp but its fine to think of it as the derivitive fix point since its the derivitive fix point) We can put a twist on that as looking for the function for which it derivitive is proportional to itself: f'(x) = a f(x) and the solution is e^(ax).
We can have higher derivitives like in the Newtonian equation of motion, we call these orders. For example F=-kx and so mx”(t) = -kx(t) and the solution is A sin(k/m t + c). This is a general solution and to get a specific one we need to state what the free parameters are, in this case A and c. We usually get them from initial or boundary conditions.
Now lets name these creatures: the mx” = -kx is a second order linear homogenous diff equation. Lets unpack! First, second, third… order means which is the highest derivitive in the equation; linear, second degree, third degree means which is the highest power in the equation so something like x’²=x would be second order. Homogenous means that there is no constant, x’-x=c would be inhomogeneous.
These are important because these properties fundamentally change a lot of things about the equation, hint methods of solution or at least tells you what mathod wont work and even determines the properties of the solution.
Order is important for the free parameters. Our second order friend had two free parameters, more accurately for the second order we get 2 free parameters per body per dimension, so a 3D grid of N balls with spring has 6N degrees of freedom if we account for them separately through dimensions.
Linearity is super important because for linear diff equations the sum of solutions are solutions, and number multiple of solutions are solutions. In fancy terms the linear combinations of solutions are solutions. So if you find a few special solutions which might be easy you can put together a complex solution with for example an infinite series. This is a common trick. Think of some rubber blanket it vibrates in some way and lets say we fix 3 edges to 0 and shake the 4th edge along some given function and we can fit to that final boundary condition usually with and infinite sum of the solutions we found.
And whether its homogenous is important because inhomogeneous diff equations are usually nasty and its good to get an early warning so you have time to mentally prepare for whats about to come.
This is great an’ all but how does one solve these equations? First of all a completely different question is whether there is a solution, and mathematicians have shown how a lot of different kinds of diff equations do have solutions and ideally we would want an equation to have exactly one solution. If those are true we can guess and check. Guess a function, take the derivatives and substitute in, if the equation is satisfied you can rest easy. But sometimes its quite hopeless to find solutions like that, for linear diff equations there are some good methods from linear algebra to algorithmicly solve them. Sometimes it makes sense to try some kind of function. Maybe look for the solution as a power series or if you have a multiple variables function like f(x,y,z) maybe look for the solution in a form like X(x) × Y(y) × Z(z), called separating the variables and its often a good trick that yields the solution. Sometimes you can even integrate with no nuance.
For example: mx” = -mg , simple free fall.
x”+g=0 and we integrate both sides
x’+gt+c1=0 and again
x+½gt²+c1t+c2=0 lets rearrange and rename the constants, you know the two free parameters we aren’t surprised to have
x(t) = -½g t² + v0 t + r0
Here v0 is the starting velocity and r0 is the starting position. We have the general solution and we can specify if we give some value for v0 and r0. We are free to put the origin at r0 so r0=0 and we pick a v0. Of course in practice we would say that x(t=0) = r0 and x'(t=0) = v0. But we can pick some other point in time as well.
We have a lot of good methods for solving these types of equations but in general they are really hard to solve, but can be applied in a lot of different cases and essentially these equations build the foundation of many branches of science like physics for instance. In their pure form however we are looking for a function that satisfies a reaction with its derivitives.
Latest Answers