What is equation: Definition and 55 Discussions

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. Danielk010

    What is the energy equation in Schrodinger's Spherical equation?

    I attempted the problem by first finding the radial, theta, and phi equation for the ground state of a hydrogen atom. I multiplied the three equations to get the wave equation. From there, I took each derivative in the Schrodinger Spherical equation and found that ## \frac {\partial^2 \psi}...
  2. bremenfallturm

    I Runge-Kutta 4 w/ some sugar on the top: How to do error approximation?

    Hello! I'm currently working with a problem which allows modelling ball motion $$\begin{aligned} m \ddot{x} & =-k_x \dot{x} \sqrt{\dot{x}^2+\dot{y}^2} \\ m \ddot{y} & =-k_y \dot{y} \sqrt{\dot{x}^2+\dot{y}^2}-m g \end{aligned}$$ Given that ##k_x, k_y=0.005##, ##m=0.01## and ##g=9.81## and when...
  3. chwala

    Find the solution to the given differential equation

    I need insight on the highlighted in Red on how ##\left[\dfrac{dz}{dx} - 1 = \dfrac{dy}{dx}\right]## otherwise the rest of the steps are clear. I just read that ##\dfrac{dx}{dy} \dfrac{dy}{dz} \dfrac{dz}{dx} =-1##
  4. chwala

    Show proof of point C in the given problem that involves Polar equation

    c Parts (a) and (b) are okay ... though the challenge was on part (a) My graph had a plot of r on the y-axis vs θ on the x-axis). The sketch of my graph looks like is shown below; I suspect the ms had θ on the x-axis vs r on the y-axis. I used the equation ##r=\sqrt{\dfrac {1}{θ^2+1}}##...
  5. chwala

    Find the equation of the invariant line through the origin

    My approach - i think similar to ms approach. The required Equation will be in the form ##y=mx## ##\begin{pmatrix} a & b^2 \\ c^2 & a \end{pmatrix} ⋅ \begin{pmatrix} k \\ mk \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} ## ##ak+b^2mk=x## ##kc^2+amk=y## ##x=k(a+b^2m)##...
  6. C

    Reduction of order for Second Order Differential Equation

    For this, I tried solving the differential equation using an alternative method. My alternative method starts at ##tv^{''} + v^{'} = 0## I substitute ##v(t) = e^{rt}## into the equation getting, ##tr^2e^{rt} + re^{rt} = 0## ##e^{rt}[tr^2 + r] = 0## ##e^{rt} = 0## or ##tr^2 + r = 0## Note that...
  7. chwala

    Solve the problem involving the velocity - time graph

    For part (a) I came up with a simultaneous equation, i.e ##m+x+4m+700## ##5m+x=700## and ##15000=\dfrac{1}{2}[5m+2x]25## ##1200=5m+2x## therefore on solving the simultaneous, ##5m+x=700## ##1200=5m+2x## we get ##x=500## and ##m=40## the ms approach is here; more less similar...
  8. mathhabibi

    I Partial Differential Equation solved using Products

    Using the concepts of Summability Calculus but generalized such that the lower bound for sums and products is also variable, we can prove that the solution to the following PDE: $$P^2\frac{\partial^2P}{\partial x\partial y}=(P^2+1)\frac{\partial P}{\partial x}\frac{\partial P}{\partial...
  9. User849483

    Need help finding uncertainty for this equation

    this is the formula v is velocity g= gravity h= height ro= outer radius of cylinder ri = inner radius of cylinder please help
  10. S

    General solution vs particular solution

    Homework Statement: What actually is the particular solution of an ODE? Relevant Equations: x Consider the differential equation ##y'' + 9y = 1/2 cos(3x)##, if we wish to solve this we should first solve the auxiliary equation ##m^2 + 9 = 0## giving us ##m=3i,-3i##, this corresponds to the...
  11. J

    Solve the commutator relations

    Hello, I need to solve the commutator relations above. I found the equation above for the last one, but I am not sure, if something similar applys to the first one. I am a little bit confused, because I know there has to be a trick and you don't solve it like other commutator. Thanks for your help!
  12. T

    I Trig Manipulations I'm Not Getting

    Hi all, I am starting with the following equation: ##2\cot\left(\frac{\theta}{2}\right) = \cot\left(\frac{k_{1}}{2}\right) - \cot\left(\frac{k_{2}}{2}\right)## with the following definitions: ##k_{1} = \frac{K}{2} + ik, k_{2} = \frac{K}{2}-ik, \theta = \pi(I_{2}-I_{1}) + iNk##, where...
  13. docnet

    Solving a first order matrix differential equation

    Let X be a continuous-time Markov chain that hops between two states ##\{1, 2\}## with rates ##\lambda, \mu>0##, so its generator is $$Q = \begin{pmatrix} -\mu & \mu\\ \lambda & -\lambda \end{pmatrix}.$$ Solve ##\pi Q = 0## for the stationary distribution, and verify that...
  14. chwala

    Solve the given trigonometry equation

    In my approach i have the following lines ##\ln (x + \sqrt{x^2+1}) = 2\ln (2+\sqrt 3)## ##\ln (x + \sqrt{x^2+1} = \ln (2+\sqrt 3)^2## ##⇒x+ \sqrt{x^2+1} =(2+\sqrt 3)^2## ##\sqrt{x^2+1}=-x +7+4\sqrt{3}## ##x^2+1 = x^2-14x-8\sqrt 3 x + 56\sqrt 3 +97## ##1 = -14x-8\sqrt 3 x + 56\sqrt 3 +97##...
  15. L

    Determine torque on a dipole and quadrupole (in external E-Field)

    For the dipole moment I calculated $$\begin{aligned} M &= \int \rho(\mathbf{r}) \mathbf{r} \times \mathbf{E}(\mathbf{r}) d^{3} \mathbf{r} \\ \mathbf{E}(\mathbf{r}) &\approx \mathbf{E}(\mathbf{0}) + \sum_{i=1}^{3} \nabla E_{i}(\mathbf{0}) \cdot \mathbf{r} \\ \mathbf{M}_{D} &= \mathbf{p} \times...
  16. jeff einstein

    B What went wrong with (-x)^2=x^2?

    I have a very basic confusion that supports some basic elements of algebra. Being a high school student my teacher couldn't answer this, hope someone could help here. We know this equation is true: (-x)^2=x^2 but once we square root both sides it becomes this: -x=x we can see this equation was...
  17. N

    B Parametric Representation of Lines

    I am having trouble with the concept that the equation L = {x + tv} is the more general form of the more familiar y = mx + b (In the first equation there should be a vector sign above the x and the v). It's hard for me to see the similarities between these two equations. 1: Even if we are...
  18. FQVBSina

    The Von Mises stress equation on wikipedia does not balance out

    On Wikipedia for Von Mises stress, it shows the following equation: But this does not work out. If I expand the second term I get: $$ \sigma_v^2 =...
  19. chwala

    Solve the given problem involving logarithms

    In my working i have, ##\dfrac{\log_{11} x }{\log_{11} 4}= \log_{11} (x+6)## ##\dfrac{\log_{11} x }{0.5781}= \log_{11} (x+6)## ##\log_{11} x = \log_{11} \left[(x+6)\right]^{0.5781}## ##x^{1.729} = x+ 6## ##x^{1.729} -x-6=0## Having ##f(x) = x^{1.729} -x-6## At this point i made use of...
  20. Clockclocle

    Homework help: Dropping a sand bag from a Hot Air Balloon

    In this situation should my free fall equation contain the v0 of the baloon or I should deny it. Because it seems to me that there is no outer force acts on the sandbag, so the scenario is just the same as I climb to the same height at time t=0 and drop the sandbag at rest.
  21. cryforhelp104

    I What does a lower case r mean over variables in an equation?

    I've attached an example here.
  22. M

    I Help in understanding Eigenvectors please

    Hi; struggling a little with eigenvectors; I can get to the equation at the foot of the example but I can't understand the "formula" leading to the setting of x = 3 at the foot of the example? thanks martyn
  23. MatinSAR

    What is ##Tds## in this equation : ##dU = Tds - PdV##

    More explanation : First law is given by ##dU=dQ+dW##. For a reversible change we have: ##dQ = Tds## ##dW = - PdV## So I rewrite first law as : ##dU=Tds - PdV## As mentioned before this ##Tds ## is the heat transferred in a reversible change. And the ##-PdV## is the work done by system in a...
  24. PhanthomJay

    B What does the Minkowski spacetime interval represent and how is it determined?

    In the Minkowski space time equation in one dimensional space , ds^2 = dx^2 - (ct)^2, what is the value to use for x and t, and what does the space time interval ds represent? For example, if Alpha Centauri is 4 light years away, what values are. used for x and t, based on speed I guess, and...
  25. T

    B Relation Between Gaussians

    Hi all, Suppose I had some some n-dimensional vectors ##\vec{a}_{1}, \vec{a}_{2}, \vec{b}_{1},\vec{b}_{2}## that satisfied ##e^{||\vec{a}_{1}||^2}+e^{||\vec{a}_{2}||^2}=e^{||\vec{b}_{1}||^2}+e^{||\vec{b}_{2}||^2}##. Now suppose there was another non-zero n-dimensional vector ##\vec{A}##. Is...
  26. L

    Show that the limit (1+z/n)^n=e^z holds

    Hi, I have problems proving task d I then started with task c and rewrote it as follows ##\lim_{n\to\infty}\sum\limits_{k=0}^{N}\Bigl( \frac{z^k}{k!} - \binom{n}{k} \frac{z^k}{n^k} \Bigr)=0 \quad \rightarrow \quad \lim_{n\to\infty}\sum\limits_{k=0}^{N} \frac{z^k}{k!} =...
  27. M

    Torque before slipping in a refrigerator

    The solution lists out mg(b/2)=ma(h/2) and then proceeds to solve for a. I am a bit stuck on how the initial equation is listed - why is the (b/2) swapped with the (h/2)? (namely, why isn't the equation mg(h/2)=ma(b/2)? My logic for this is y-direction and x-direction ) I feel that I am missing...
  28. MatinSAR

    Find ##b-a## which satisfies following limit

    ## \lim_{x \rightarrow 1} {\frac {x-2} {x^3+ax+b}} = -\infty## The limit is equal to ##\frac {-1} {1+a+b}## . so I can say that ## a+b = -1 ##. But I cannot find another equation to find both ##b-a##.
  29. MichPod

    A negative number to a fractional power -- Is it legit in High school?

    My son (11th grade, Canada school) brought an equation ##x^\frac{3}{5}=\frac{x}{4}## from his class on which the teacher says it has a ##x=-32## root, in addition to ##x=0## and ##x=32##, of course. That was a bit a surprise for me as I was taught in my school time that only a non-negative...
  30. J

    1999 AP Physics C Mech: Conservation of momentum and energy

    Why is (1/2)(mv0)^2 = 1/2(M+m0)gh not a valid equation for conservation of energy? Isn't the energy from when the dart is shot the same as when the two masses move at speed v?
  31. C

    Find the equation of state of each gas

    The problem is from the book "The Principles of Thermodynamics" by ND Hari dass. It looks trivial problem, but I am not able to form logical arguements for going into next step. For example, It seems like first gas has equation of state ##PV =nRT## and second has ## \left( P_2 +\frac{a}{V_2^2}...
  32. D

    Find the relation between 2 variables

    Here is the equation I obtain after simplification, I don't know if it is correct: gmc * V1 + s * C2 * Vout = [{s * (C1 + C2) * ro2 + 1} * Vout - s * C1 * ro2 * V1] * (s * rb * C2 + 1) / {ro2 * rb * (s * C2 - gm2)} I need to eliminate V1 to find the relation between Vin and Vout.
  33. chwala

    Find the value of ##a, b## and ##k## in the problem involving graphs

    In my approach i have the roots of the equation being ##x=a## and ##x=b##. There are two assumptions, In the first assumption, ##a=\dfrac{1}{2}b## ##2a=b## then, ##4=k(-a)^2(-2a)## ##4=-2ka^3## ##⇒ -2=ka^3## Now since ##2a=b## then ##a=1, b=2⇒k=-2##. our equation becomes...
  34. neilparker62

    I Question on Lambert W function

    In the following I ask WA to solve the given equation and it produces a solution using the Lambert W function. I thought : $$W(x*e^x) = x$$ but here it seems $$W_n \left(\frac{-MT}{P}*e^{\frac{-MT}{P}}\right) \neq \frac{-MT}{P}$$ Is there a difference between ##W(x)## and ##W_n(x)## ?
  35. ikihi

    Calculus Help locating a Math Equation to fit this curve

    The concentration is the Y axis, and the values are the x axis. What is the equation for finding a target value on a curve between set of points?
  36. J

    Kinematics: 100 meter dash when one runner starts 3 seconds later

    $$t_A= 3s$$ $$ s= 100m$$ $$v_A= 5m/s$$ $$v_B=7m/s$$ $$s(t)=7(t-3)+0$$
  37. chwala

    Solve the equation : ##x^4+12x^3+2x^2+25=0##

    Wolfram gave the solution and a hint: i want to understand the hands on approach steps... In my approach (following Wolfram's equation) i have, ##(x-3)^2(2+12(x-3)+(x-3)^2=-25## ##(x-3)^2((x+3)^2-33)=-25## ##(x-3)\sqrt{((x+3)^2-33)}=-5i## ...
  38. H

    If tension provides the torque to this pulley, what is its equation?

    15 - Tension = 1.53*a T = 15 - 1.53*R*α T*0.33 - 1.10 = Iα Is this approach correct?
  39. A

    I Adiabatic Processes: Why does this equation use constant volume and not constant pressure?

    Why is this equation (red sign) written in constant volume and not in constant pressure?
  40. PLAGUE

    I Why do the lines m and l coincide in the proof of the parallelogram rule?

    I was going through this book called "A Course in Mathematics for Students of Physics Volume 1 by Paul Bamberg and Shlomo Sternberg". There in a part they said something like this: ...if we start with a point P and write ##R=P+u## ##Q=P+v## and ##S=P+(u+v)## then the four points ##P,Q,S,R## lie...
  41. chwala

    Find the value of ##k^2## in the problem involving trigonometry

    In my working i have, ... ##\cos C = 2\cos^2 \dfrac{1}{2} C -1## ##c^2= a^2+b^2-2ab(2\cos^2 \dfrac{1}{2} C-1)## ##c^2= a^2+b^2+2ab(1-2\cos^2 \dfrac{1}{2} C)## ##c^2= (a+b)^2 (1-2\cos^2 \dfrac{1}{2} C)## Now from here, ##k^2 =2## but text gives different solution. I am still checking...
  42. lukegxhycj

    SUVAT equation question: When will the car collide with the lorry?

    s1=-u1^2/2a1 s2=-u2^2/2a2 s2>s1+d (If distance the car stops is bigger than the distance the lorry stops plus the initial distance then they will crash) (sub s1 and s2 in) -u2^2/2a2 > -u1^2/2a1+d Switch 2a2 with whole left side of equation. -u2^2/(-u1^2/2a1)+d > 2a2 Make the (d)a fraction by...
  43. chwala

    Solve the equation: ##\tan x ⋅\tan 4x = 1##

    I saw this link; is the approach here correct...
  44. T

    MTW Box 21.1 - What can "add and subtract" do for Equation (12)?

    I haven't posted for a while and I am still (!) working through some of the things I didn't quite get in MTW Chapter 21. Here is my latest puzzle. I want to work out how to get from Equation (12) in the attachment, to Equation (15). I've tried the "add and subtract" ##\{\frac...
  45. L

    I Finding the equation of a curved line

    Ok, so i have done many math classes, as i am an engineer, however, a theory class and proof class i h ave not done, except maybe one haha. here is my question, if i draw some sort of curved line by hand, how do i find an equation for that? for example, i draw half of a tear drop but along...
  46. W

    Attempting to teach this: Momentum and Impulse

    For question 1. I am stuck. I know that the equation involves time and possibly rate, should solve for distance. But not sure how to set it up with information given. 2. Ft= m 🔺️ v F(3)= (100kg)(30m/s) 3 s= 3000 kg m/s Same applied to question 3. 3. F(2)= (100kg)(-30m/s) F(2) = -3000 kg m/s...
  47. FlowiwGhar

    Physics problem - Salmon jumping waterfall

    m * g * h + (1/2) * m * v² = m * g * y Simplifying the equation: g * h + (1/2) * v² = g * y Substituting the values: g * 0.614 + (1/2) * v² = g * 2.73 * sin(26.7°) Now, let's solve for v: (1/2) * v² = g * 2.73 * sin(26.7°) - g * 0.614 v² = 2 * (g * 2.73 * sin(26.7°) - g * 0.614) v = √(2...
  48. al4n

    B How to find an integer solution to a nonlinear equation?

    given something like: an = c where c is given and a, n, and c are only allowed to be integers. how would one find the value of say n or a?
  49. H

    A Proving an equation in a book about Periodic Structures

    Hello everyone, I am reading some book titled: Periodic Structures: Mode-Matching Approach and Applications in Electromagnetic Engineering. In Chapter 2, there is an equation as follows: where . Here the electric field is along the transverse x − y plane like the propagation vector kt. Now it...
  50. jamesbrazil

    Maple Finding the solution of this long equation using Maple

    I need help solving an equation. I started using Maple, but had no success. Could someone explain to me which command to use? I need to find a very small value of ##x##, that is, ##x \ll 1##. The equation is $$434972871000000000.0+{\frac {\sqrt {6} \left( { 1.488388992\times 10^ {-36}}\,\ln...
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