If you were to solve this equation, you would start with a general solution and from there get a more specific solution, in this case a good starting point would be y(x) = Ce^(Ax), where A and C would be constants that you try to limit by inserting this general solution on the differential equation.Īfter doing that you will find that A can be either 1 or -3 for this particular differential equation ( y'' + 2y' - 3y = 0) so the general solution becomes y(x) = C₁eˣ + C₂e⁻³ˣ, where C₁ and C₂ are only limited by initial conditions (not given in this problem, so you can choose whatever values you want and it will work). Since the derivatives are only multiplied by a constant, the solution must be a function that remains almost the same under differentiation, and eˣ is a prime example of such a function. The differential equation y'' + ay' + by = 0 is a known differential equation called "second-order constant coefficient linear differential equation". It is up to you to make decisions to determine what the answer is at the end of the day.Įdit: This question was so good, I decided to write a more in-depth answer to it on my blog (where I have the freedom to use mathematical notation and add pictures. And let this be a lesson: always be careful what you plug into your calculator, because your calculator will just blindly calculate numbers. I'm upvoting your question because it's a great question, and very well observed. While you might not think it is, when dealing with complex numbers, this is the principal root and it's a little too difficult to explain why without drawings. So, in the case here, what the calculators are giving you called the "principal root". All it specifies is that there will always be three different roots. So, if you're interested in cube roots of a number (as you are in this example), you will always have three roots.
When you take the nth root of a number, there are always n different roots In essence, the takeaway from it in the context of this problem is as follows: The reason comes from the Fundamental Theorem of Algebra. The 1 calculator that I found that showed a real solution was Symbolab.
I know this is also the case for -5/3, try that on paper too. If you have any insight on the weirdness, please give me some insight below! Thanks for your time! :D I asked my math teacher and he said it should be real. But when you type in (-7/3)^(-7/3) onto most calculators, google included: I was finding y when x = -7/3 when I noticed something strange. I was using pencil and paper for most of my problems to determine whether the y values (given x) were real or not. Then I started to find as many points I could in the negative section to see if there was a pattern. This lead me to wonder whether there are points in between the integers, and there are! Such as when x = -1/3, yet not all are real numbers.
I recognized that some values are complex and cannot be graphed, yet some can, such as (-1, -1) or (-2, 0.25). I noticed the function abruptly stopped at the y-axis and did not extend into the second or third quatrains at all. I'm not exactly good at math so forgive me if this question seems really silly.Ībout a week ago, I was wondering in my math class what the graph x^x looks like, and since we were using the desmos calculator in class, I tried it out. Don't have a question about this video, but I don't really have a good category to put it.
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