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
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.
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