In the previous two chapters, we dealt with two specific topics on derivatives and integrals. We took this opportunity to look at some concrete examples, but it was not conceptually revolutionary - we imagine that now. In fact, we will continue our trip to the differential analysis with a field trip to the differential equations. These equations combine both derivative and integration tools, so we're probably going to sweat a little, but for that we can apply our knowledge to hitherto elusive problems.
How different a tool are differential equations from what we've encountered so far? Derivation and integration allowed us to look at a function and understand it a little better. It was useful for calculating its average value, or for calculating its maximum. That's great if we have a problem and can describe it with a familiar function, but that rarely happens in the real world.
In contrast, differential equations can tell us what function to use at all. It's like we can add up numbers, and so we can tell how much our purchase costs, but that's it. But these basic capabilities will expand dramatically if we learn the equations: we can learn how many things we can buy, or how much they are supposed to cost.
The solution to one particular, so-called Navier-Stokes equation is one of the problems of the millennium -- mathematical problems for which the solver faces a $1 million reward.
But differential equations are much deeper than ordinary equations. They don't normally have just one solution, we don't often know the solution, or it's some very exotic function. We often find a new function by solving a certain equation. Furthermore, we can also extend differential equations into multiple dimensions, into strange mathematical spaces, or provide various other mathematical tinkers (e.g., metric space), but later on. It's really important not to despair here! If differential equations were simple, they wouldn't be worth studying because they couldn't describe complex problems.
The normal equation that every grade-schooler encounters, we can express with the popular children's game I Think Number... The result of this game is solving the equation and getting a number. On the other hand, the differential equation is a type of problem I think a function... Moreover, there are derivatives in its formulation, so the result is often unexpected.
Try to think about whether you can invent and solve some simple (even trivial) differential equation.
For example, the following differential equation is a simple example: I think of the function of the variable $x$. If I do its derivative, I get a function of $x$. What was my function? When we mark the search function $f(x)$, we can write the equation mathematically: $$f'(x) = x \,,$$
$$\begin{align*} \int_0^X f'(x) \, \mathrm{d} x = \int_0^X x \,\mathrm{d} x = \left[ \frac{1}{2} X^2 \right]_0^X \\ f(X) = \frac{1}{2} X^2 \,. \end{align*}$$
This ambiguity is not unique to differential equations. For example, a problem I think of a number, if I multiply it by itself, I get 64, what's the number? has a solution of 8, but also -8. All differential equations are so ambiguous, which may seem annoying. Perhaps we'll learn later that it's actually useful.
We figured out the $x$ integral simply because we know the primitive function to $x$, you'll also find it in the Derivative Appendix. So we present as a result the function $f(x)=x^2/2$, indeed its derivative is equal to $x$. However, this function is not the only solution to our problem, e.g. $g(x)=x^2/2 +1$ is also equal to $x$ after the derivative. We see that adding any constant to the result gives us another solution. This is because in integration we could start to integrate not from scratch, but from any other point. This would result in the result differing by a constant. The more correct course of action would be to integrate not from zero, but from some point of $C$ constant that we will determine later. Alternatively, we can write the result as $$f(X) = \frac{1}{2} X^2 +C \,.$$
We can sum up our findings: if we integrate a function in a differential equation, it must be within indeterminate limits. As a result, we get another function of an independent variable, and we add a constant to it to describe that uncertainty. For simplicity's sake, we even skip the integration limits, saying that we integrate vaguely. So we can rewrite the differential equation above using the new notation: $$\begin{align*} f'(x) &= x \\ \int f'(x) \, \mathrm{d} x &= \int x\, \mathrm{d} x \\ f(x)&= \frac{1}{2} x^2+C\,. \end{align*}$$
Our example of a differential equation may have seemed straightforward, but we can't go backwards in every equation. This is what we see in quadratic equations-to deal with them, we have to know the formula for solving them. In the case of differential equations, the situation is even more complicated, as there is usually no formula to solve them.
If the derivative was squeezing the tube out of the paste and integrating inverse surgery, then the solution to differential equations is building sculptures out of toothpaste. The two previous skills will no doubt come in handy, but it takes more than craftsmanship to work effectively. Different methods are used to solve these equations, and we still can't solve some differential equations. On the other hand, the difficulty of solving these equations makes them interesting, if they were simple, we wouldn't be able to use them to explain the complex things we often see in life.
One of the main tasks of naked algebra is to present the basic kinds of differential equations and understand their solutions. But an equally difficult task is to find and articulate the differential equation in a given problem. Sometimes the task is harder than solving it (after all, we can always use machines and calculators to solve it). So that's what we're going to focus on.
Differential equations are good to demonstrate in practice. So we look at the famous example of Newton's laws, which describe the movement of material bodies. These are well-known school laws, and they also describe what happens in everyday life. So you can easily imagine the procedure in their example.
Isaac Newton has formulated a total of three laws that describe a general method for dealing with the movement of material points in the presence of forces. Newton uses these laws to construct a differential equation, and its solution describes the movement of the bodies-because of the great importance of this equation and illustration, we will now focus on Newton's laws. The first reads as follows:
Bodies remain in a state of calm or even straightforward motion when no force is exerted on them, or the result of forces exerted on them is zero.
This law says that the primal or fundamental state of affairs is either calm or even straightforward movement. We can understand the calm, it makes sense why this would be the natural state of the body. But if we imagine, for example, a speeding car, we had to give it some energy to get going. Indeed, with a car, we can say that it has some quality that separates it from the stationary body: a mention of kinetic energy, but perhaps speed. The catch is precisely this speed: how do we define it?
From the driver's perspective, the car is stationary and everyone around it is moving at high speed. If a car driver measured his speed, he would measure zero as well as kinetic energy. That's why we have to define what speed means first.
In Newton's terminology, there are so-called reference systems, i.e., worlds from someone's point of view. For example, there is a reference system from the point of view of the centre of the Earth, the centre of the Sun or any human. In fact, Newton's first law between the lines says that there is an inertial reference system, i.e. one that is at rest. There are other inertial systems that differ from the original only because they move at some fixed speed. For example, a car system and a road system are inertial to each other. But if the car is taking off from the road, it's not inertial relative to the road because it's accelerating.
So it is natural to describe the world from the point of view of an inertial reference system, and it can contain either bodies at rest, in uniform straightforward motion, or accelerating. But bodies in uniform straightforward motion are at rest from the point of view of some other inertial system-that's why they're also in their natural state. It's those accelerating bodies that aren't in basic condition. And her description dwells on Newton's second law:
Force, momentum, etc. are of course vector quantities, but let's not consider that for the sake of simplicity.
Bodies can be attributed a momentum of $p$ equal to the product of their mass and speed: $p=mv$. The force of $F$ exerted on a body causes it to change its momentum (change in speed over time). Written by equation: $$\begin{align*} F &= \frac{\mathrm{d}p}{\mathrm{d}t} = \frac{\mathrm{d}(mv)}{\mathrm{d}t}= \frac{\mathrm{d}m}{\mathrm{d}t}v + m\frac{\mathrm{d}v}{\mathrm{d}t} \\ F &= \frac{\mathrm{d}m}{\mathrm{d}t}v + m\frac{\mathrm{d}v}{\mathrm{d}t} \,. \end{align*}$$
Here we used the product derivative rule. In case the mass of our observed bodies does not change (this is almost always the case), we write $\mathrm{d} m /\mathrm{d}t = 0$. Next, we mark the speed change in time as acceleration: $a\equiv \mathrm{d}v/\mathrm{t}$. We can now formulate a more familiar form of the Second Newton Law:
A more general version of Newton's Law will still come in the Ciolkovsky Missile Equation example.
The $a$ acceleration caused is directly proportional to the $F$ force exerted and inversely proportional to the $m$ body weight. Written by equation: $$F = m \cdot a \,.$$
It's easy to predict how a uniformly straightforward moving body will evolve. It's more difficult with accelerating bodies, and the equation above gives us a clue. Let's imagine this with a simple example of a $m$ stone falling in a gravity field with a gravity acceleration of $g$. Gravitational force here is $F=mg$. We then get an equation that we call the movement equation: $$\begin{align*} F &= ma \\ mg &= ma \\ g &= a \,. \end{align*}$$
But the simple $g=a$ equation is really a simple differential equation in disguise. Because acceleration is a change in velocity over time. Symbolic: $$a = \frac{\mathrm{d} v}{\mathrm{d} t} \,.$$
Next, speed is the change in position over time. We get that $$a = \frac{\mathrm{d}^2 x}{\mathrm{d} t^2} \,.$$
Let's add to the equation: $$g = \frac{\mathrm{d}^2 x}{\mathrm{d} t^2} \,.$$
We integrate the equation twice: $$\begin{align*} \int g \, \mathrm{d} t= \int \frac{\mathrm{d}^2 x}{\mathrm{d} t^2} \, \mathrm{d} t\\ gT + C= \frac{\mathrm{d} x}{\mathrm{d} t}\\ \int (gT+ C) \, \mathrm{d} t= \int\frac{\mathrm{d} x}{\mathrm{d} t} \, \mathrm{d} t\\ \frac{1}{2} gT^2+ CT + K = x\,. \end{align*}$$
For clarity, if we mark $T$ as $t$, the $C$ integration constant as $v_0$, and the $K$ constant as $x_0$, and swap sides of the equation, we get: $$ x(t) = x_0 + v_0\cdot t + \frac{1}{2} g t^2 \,.$$
It gave us a familiar formula from elementary school for accelerated motion in the gravitational field. The integration constants further guaranteed that it was valid for a wide range of situations, depending on what initial speed or position the falling stone had. We also see that we integrated twice, so we have two integration constants.
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