equations Definition and 62 Threads

In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign =. The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation.Solving an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables.The "=" symbol, which appears in every equation, was invented in 1557 by Robert Recorde, who considered that nothing could be more equal than parallel straight lines with the same length.

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  1. J

    Calculating the sum of a series of cubes

    Mentor note: Moved from technical math section, so missing the homework template filling in the boxes is easy, 1,3,6,10,15 second is just squares of that, 1,9,36,100,225 But how I anyone supposed to find and expression for this? This is from a textbook on elementary algebra, the specific...
  2. P

    Torque on a sign hung by a hinge

    Since I understood that torque = net moment, and as it said angle from the vertical, I used the equation t = F*d*cosx. So, 6 = 40*0.4*cosx, x = 68 degrees. I got a zero out of 5 in this question because the thought process was wrong as well as the answer. I do not understand this problem beyond...
  3. P

    How to approach this question using SUVAT equations?

    Hello, I came across the question attached. To approach this, I assigned S1 to car Y and S2 to car X. So the displacement of Car Y is S1=S2 + d(naught). Then using S1= ut+(1/2) at^2 for car Y and S2= uT , I got to uT + d(naught) = VT + (1/2)aT^2. However, I am stuck here as I can't relate...
  4. J

    Confused about kinematic equations

    x=x0=vt v=v0+at x=x0+v0t+(1/2)at2 That (1/2) there does not make sense to me. It makes sense mathematically, if you integrate the velocity formula you get the (1/2) as a result of the integration rules and if you differentiate its needed to cancel out the 2 and arrive at the velocity...
  5. chwala

    Use Wronskian method in solving the given second order differential equation

    I am looking at this link; https://tutorial.math.lamar.edu/Classes/DE/RealRoots.aspx ##y{''} +11y{'} +24 y=0, y(0)=0, y^{'}(0)=-7## Now the general approach of applying boundary conditions directly is quite straightforward to me. I am interested in using an alternative approach, that is the...
  6. A

    Explanations On Entering Various Equations Into Mathematica?

    I've had Mathematica for a long time, but up until now, I just used to use the free-form input. I'm trying to learn the language so that I'm able to do more with the software. However, the information I gather from the Wolfram site is worthless for a lot of things, as I don't understand what...
  7. A

    Mathematica 3D Graphing (3D Plot) Issue

    I'm trying to graph two equations to compare the two visually. For some reason, one of them isn't showing up, even though I'm in a range to where it should be. See the screenshot below. The "3x - y = 4" is graphing fine, but for some reason, "x + y + z = 5" isn't showing up. At first, I...
  8. T

    Problem with dimensionless quantities in an equation requiring M^2 (Super Hamiltonian Formulation for Geodesic Motion)

    I've worked my way through this exercise but I am a bit puzzled by the last line "What do the choices ##\mathcal H = -½, 0, -½\mu^2, +½ ## mean for the geodesic and the choice of parameterisation?" I've worked to produce: ##\big (\frac {dr}{d\lambda} \big ) ^2 = E^2 - \big (1- \frac {2M}{r^2}...
  9. T

    Engineering Determine the flight time and heights of 2 balls thrown vertically

    To find the time of this problem, I used the equations 3 + (10 * x ) = 13.5 *x. 3 = 3.5*x. Therefore x was 0.857. To start off with child A was already three feet above ground and I got the answer to the time. However I couldn't find the height.
  10. U

    I Claude Naiver work on Navier-Stokes equations

    I am interested what are main differences/news in two Claude Navier works "Sur les lois des mouvements des fluides, en ayant égard à l’adhésion des molecules"- memoire 1 and "Sur Les Lois du Mouvement des Fluides" memoire-2? Are both works written(not published) in 1821? Does memoire 1...
  11. chwala

    Find the possible values for ##a## and ##b## where ##a## and ##b## are integers

    I noted that, ##lcm(a,b)=2048## Letting ##a=2^x## and ##b=2^y##, ##⇒\dfrac{1}{2^x}+\dfrac{1}{2^y}= \dfrac{3}{2^{11}}## ##⇒\dfrac{2^{11}}{2^x}+\dfrac{2^{11}}{2^y}= 3=[4-1]## ##⇒\dfrac{2^{11}}{2^x}+\dfrac{2^{11}}{2^y}=2^2-2^0## My intention being to write all the numbers to base ##2##...
  12. W

    Diophantine equations and physics

    Hello, can anyone tell me about applications of Diophantine linear equations to physics?The only example that comes to mind is that of a series of weights. I have a measuring rod that is 40cm long and hinged in the middle. If I put a 40g weight at one end, how can I place 5g, 7g, 11g, 13g, 17g...
  13. P

    Correct Derivation of the Adiabatic Condition? (PV Diagram)

    By the First Law, Definition of an Adiabatic Process, and Definition of Work: ##\Delta E = Q - W = - W = - P \Delta V ## (because ##Q = 0##) (Equation 1) By the Equipartition Theorem: ##\Delta E = \frac{3}{2} Nk \Delta T ## (Equation 2) By Setting Equation 1 equal to Equation 2 ## \Delta T...
  14. lebronJames24

    Magnus Effect Equations and Mechanics

    Ok I need to clear up a few things. First, in (Figure 1), a soccer ball (the circle in the photo), would have the magnus force towards the lower pressure because the universe favors high entropy? Also what are the forces are active in figure 2? Lastly what to the variables in Figure 3 mean...
  15. M

    Solving inexact ODE

    For this problem, Does someone please know how they combined the two equations for H to get the finial equation ##H(x,y) = .... = c##? Thanks!
  16. M

    Coupled system of differential equations

    For this problem, Would it not make sense to use for ##x' = -sx - gx - ry## as a better version of ##x' = - gx - ry## since the ##sx## term connects the two DEs to form a coupled system (from what the author explains the ##sx## term represent insulin glucose transformation). Thanks!
  17. M

    Solving system of differential equations using matrix exponential

    For this problem, The solution is, However, can someone please explain to me where they got the orange coefficient matrix from?It seems different to the original system of the form ##\vec x' = A\vec x## which is confusing me. Thanks!
  18. M

    Complex Eigenvalues of system of DE

    For this problem, Can someone please explain to me how they got from the orange step to the yellow step? I am confused how the two expressions are equivalent. Thanks!
  19. alejandromeira

    Praise Issues with LaTex

    Hello, can anyone give me a hand? I am trying to make a post but the LaTex engine doesn't work properly. I Try the `four dollars´ and the `four hashtag´ method on, but the equations dont appear when I click the Preview button. The LaTex engine doesn't do anything. I have posted several threads...
  20. H

    A Equations of Motion for Torsional Inverted Pendulum on a moving cart

    Hello I am trying to figure out the equations of motion for the following scenario: a mass m is held upright on by a massless rod mounted on a cart that is being pulled by a linear force. The caveat is that the rod is mounted on the cart via a torsional spring which causes a restorative torque...
  21. DylanMurfly

    Cables, pulleys, 2 weights

    Fx = 0 = ABcos(a)+BCcos(b)-BDcos(30) Fy = 0 = ABsin(a)+BCsin(b)-BDsin(30) == Fx = 0 = ABcos(a) + 3.2737cos(b) - 3.738 Fy = 0 = ABsin(a) + 3.237sin(b) - 2.158 But i cant find a third equation to use. I've tried a+b = 90 but that produced a number of errors. Thank you. Edit: angle b is the...
  22. chwala

    Show that the system of equations has no unique solution

    solution; @Mark44 we discussed this sometime back on the forum. The approach of using determinant or echelon form of a matrix was sufficient. Would i be correct to use this approach, ##x+2y+3z=4x+5y+6z## ##3x+3y+3z=0## and ##7x+8y+9z=x+2y+3z## ##6x+6y+6z=0## Then, it follows that the...
  23. person123

    My Website For Creating Interactive Visuals Linked To Equations

    I recently finished a website and I'd like to share it in case it would be useful, as well as to get feedback. The URL: equationvisuals.com The website uses a basic computer algebra system (CAS) I built to solve systems of equations and show its steps. These equations can then be linked to...
  24. brotherbobby

    LaTeX Possible name of the software used to type equations in a video

    Video Link Many thanks if you can guess what the name of the software is. It can be very useful for teaching. Let me also confess that, till date, I haven't found a software that can type ##\rm{LaTeX}## equations in real time on to the computer screen. LyX is the only one I know of, but the...
  25. L

    I Lorentz Transformations Confusion

    Two simple equations, yet I can't find any sources that explains it very well. If you have any suggestions on where I could find this then please link them! $$ \Delta{t'}=\gamma(\Delta{t}-\frac{v}{c^2}\Delta{x}) $$ Does it matter if S' is moving away to the right, or to the left, of S? I...
  26. billtodd

    The ratio between two horizontal speeds in this pendulum arrangement

    From the forces equation I can only understand from it that the forces' equations are:##N=Mg## and ##T\sin \theta=m\ell \ddot{\theta}##. But I don't know how to find the Torques' equations. Any help is appreciated. N=Mg ##Tcos⁡θ+N=mg
  27. G

    I Why Do Some Theorists Claim Cosmic Inflation Dilutes Magnetic Monopoles?

    So if Maxwell's Equations state that Div B = 0 which is a statement that there are no magnetic monopoles why do some folks claim that Cosmic Inflation "dilutes" magnetic monopoles?
  28. chwala

    Work out the amount that Arjun paid in rent in 2019

    I am interested in a more direct linear equations approach...to solve, which i know is possible... A. My initial thinking was along the lines, let ##x## be the amount that Arjun paid in 2018 and let ##y## be the amount that Gretal paid in 2018 where A was the total amount paid in 2018 ...
  29. Safinaz

    Simpliying this partial differential equation

    Does this equation mean that: ## x+y =z ##, and ## x = y##? I mean ## \delta_{ij} ## terms in the LHS of the eqaution equal those at the RHS ? with knowing that ## \partial_i \partial_j x ## term dose not vanish for ## \delta_{ij}=1 ## Any help is appreciated!
  30. A

    Solve for all angles x: cos(2x) + cos(x) = 0, where 0<x<2pi

    I'm not sure how to go about solving this mathematically? In just using what seems obvious, I know the angle pi would work, because pi = -1, and 2pi = 1. However, as far as manipulating the equations in a way where it can solve itself without me having to look at a chart where cos for both x...
  31. T

    I Heisenberg Equations of Motion for Electron in EM-field

    Consider the Heisenberg picture Hamiltonian $$H(t) = \int_{\textbf{r}}\psi^{\dagger}(\textbf{r},t)\frac{(-i\hbar\nabla+e\textbf{A})^{2}}{2m}\psi(\textbf{r},t)$$ where ##\psi(\textbf{r},t)## is a fermion field operator. To find the equations of motion that ##\psi,\psi^{\dagger}## obey. I would...
  32. chwala

    I Solve the system of linear equations

    This is fine but i have my own approach as follows; ##\begin{pmatrix} -7 & -6 & -12 & -33 \\ 5 & 5 & 7& 24 \\ 1 & 0 & 4 & 5 \end{pmatrix} ## → Row 1 times 5 and row 2 times 7... ##\begin{pmatrix} -35 & -30 & -60 & -165 \\ 35 & 35 & 49& 168 \\ 1 & 0 & 4 & 5 \end{pmatrix}## →...
  33. paulb203

    Basic Simultaneous Equations Problem

    I got c=13, but I'm using an app that, at the 'diagnostic' stage, doesn't tell you if you're correct or not (I multiplied the first equation by 4, and the second by 2, then subtracted the second from the first)
  34. B

    Derivation for modified sine curve equations

    Hello, I am looking for a detailed derivation of the equations used to generate the modified sine curve. I found one in Cam design handbook by Harold A. Rothbart but I didn't understand how we get certain equations. My end goal is to combine the modified sine curve with constant velocity and get...
  35. Filip Larsen

    I Solving matrix commutator equations?

    I have the matrix relationship $$C = A^{-1} B^{-1} A B$$ I want to solve for ##A##, where ##A, B, C## are 4x4 homogeneous matrices, e.g. for ##A## the structure is $$A = \begin{pmatrix} R_A & \delta_A \\ 0 & 1 \end{pmatrix}, A^{-1} =\begin{pmatrix} R_A^\intercal & -R_A^\intercal\delta_A \\ 0 & 1...
  36. MatinSAR

    Find thermal capacity of a Van der Waals gas

    Can anyone guide me how can I find ##C_P## and ##C_V##? There are two equations in my book which I think can help but I'm not sure if I can use them to find ##C_P## and ##C_V## of a vander waals gas. I know equation of state of this gas.
  37. W

    Troubleshooting Circular Motion: Solving the Toy Car Loop Puzzle

    I tried using conservation of energy, and using the equations for circular motion, but I can't seem to find a solution. Any help?
  38. S

    B More examples of equations that unexpectedly model nature in similar ways

    Been dipping my toes into maths by examining how equations work on the most basic level, and I love encountering equations that turn out to model similar aspects in nature, for example the inverse square law is apparent in equations for gravity and for electromagnetism. In the thumbnail of...
  39. L

    B Which equation should be used to calculate impulse: I=F(t) or I=m1v1-m0v0?

    Just going over some basic equations to help someone out and i have noticed that i get 2 different answers when using 2 different but very similar equations for impulse. first eq. I=F*t lets say that i have a small rocket motor that has a force of 5.95 newtons and a impulse of 2.5 n-s...
  40. Ben2

    Show that the Poiseuille field of flow is rotational

    I tried using these equations, but it's not clear if we should hold y_1 = y_2. A transverse velocity vector would produce a flow at some angle to the horizontal, but How do they known there's such a vector?
  41. S

    A Exploring the Restricted Three Body Problem: Hamiltonian vs Lagrangian Approach

    I am having a difficult time finding papers that derive the equations of motions of the Restricted Three Body Problem using Newtonian mechanics. I have found some (barely) for the CRTBP, but there are none on the ERTBP. Does it depend on what is being studied within the problem that the...
  42. P

    A Quintic and Higher Degree Polynomial Equations

    What's the root formula for fifth and higher degree polynomial equations, which have roots in radicals?
  43. W

    I The electric field due to a dipole

    and Can someone explain how they made these equations like this? How did the radius become that equation? What formulas from algebra did they applied? I'm looking at these formulas and I don't understand how r=z+1/2*d
  44. E

    Why is the square root of x^2 = |x|?

    If I reason this as follows, I run into problems. Please help me understand what is wrong with reasoning like this. a) I start with the left hand side of the equation and let that x be -2. b) I square it. This gives me 4. So I now have the square root of 4. c) The square root of 4 is +/- 2. The...
  45. L

    Show that these equations are linearly dependent using Mathematica

    TL;DR Summary: solve linear systems of equations Hi, I am supposed to solve the following problem with Mathematica This is the tutorial we received on how to solve linear systems with Mathematica: I then tried the whole thing for the task, but unfortunately Mathematica doesn't give me any...
  46. M

    B Solving Simultaneous equations with cos() and sin()

    Hello I have two equations given below and answer is also given solve by simultaneously i am not sure how this happen please guide thank you
  47. brotherbobby

    LaTeX 1. Colour while using ##\LaTeX## 2. Preview first time

    I have observed that one cannot use colour here while typing something in ##\LaTeX##. Let me try : ##\text{Let's see if this can be \color{blue} {coloured}?}##. Nope, I type, via an edit after posting. Of course I can use colour from the panel above, but that will have to work for the entire...
  48. barryj

    I Using an array to solve a system of equations

    I was given a problem to use an array, not a matrix, to solve a system of equations. I have not heard of doing this and can not find any reference to how to do it. like... 2x+3y = 12, and 3x-4y = 14 How is this solved using an array or grid?
  49. K

    Finding value of g given a motion map

    I tried using all equations of motion but couldn't get the correct answer. Any hint would greatly help. I've tried doing this for two days now!
  50. J

    B Understanding the Relationship Between i*cos and sin in Circuit Analysis

    In circuit analysis, everything seems to work out when you set i*cos = sin. But thats not a legitimate equation, so why does that work? Is there a proof that this is a real equation?
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