equation Definition and 101 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. M

    How should I show that this root is given approximately by this?

    Proof: Consider the equation of ## a^2-x^2=\epsilon\sinh x ## for ## 0<\epsilon<<1 ##. Let ## \epsilon=0 ##. Then the unperturbed equation is ## a^2-x^2=0 ##. This gives ## a^2-x^2=0\implies (a+x)(a-x)=0\implies x=\pm a ## with the root ## x=x_{0} ## such that ## x_{0}(a)=a ## because ## x\geq...
  2. M

    How should I show that all solutions of this equation are bounded?

    a) Proof: By definition, the potential energy ## V(x) ## is given by ## F(x)=-\frac{dV}{dx} ##. Note that ## \ddot{x}=-\frac{dV}{dx} ## where ## \ddot{x}=-x-\epsilon(\alpha x^2\operatorname{sgn}(x)+\beta x^{3}) ##. This gives ## \frac{dV}{dx}=x+\epsilon(\alpha x^2\operatorname{sgn}(x)+\beta...
  3. S

    How do I solve the Laplace equation using Robbin's Boundary Conditions?

    I've tried a few things. I did one method to try to accomplish the removal of the -70 in the derivative boundary condition. It came out as below. When plotting it however it gave a solution that didn't make sense.
  4. DaveC426913

    B Is this equation at all sensical?

    (Yes, I'm guilty of passing along something incomplete and of utterly unknown provenance that someone saw somewhere and didn't understand. I feel dirty.) ∀t(T(x,t)⟹(¬H(x,t)∨H(x,t+Δt))) Certainly, without specifying any of the variables, it's got to be useless. At least we can assume t and Δt...
  5. chwala

    Find h in the problem involving a stone thrown upwards

    Was the approach used by the referenced tutor correct? In my understanding, the set up equation ought to be, ##s=-h, u=16## as the stone is being thrown upwards and ##g=-9.8## giving me, ##-h = 64 - \dfrac{1}{2}×9.8 ×16## ##-h=-14.4##m ##h=14.4##m , or it does not really matter.
  6. L

    Find all positive integers n such that 3x^2-y^2=2018^n has an integer solution

    I need an idea. Thank you.
  7. M

    I Question on interference equation in a recent paper

    The paper follows (free): https://arxiv.org/abs/2111.03203v2 I was trying to read it, but couldn't understand where equation 2 comes from (page 3). Is this just a standard diffraction or interference equation? Could anyone provide a reference that explains that equation?
  8. L

    A number Theory Question: Solve 2^x=3^y+509 over positive integers

    My attempt and solution : $$2^x=3^y+509\Longrightarrow 2^x-512=3^y+509-512\Longrightarrow 2^x-2^9=3^y-3$$ $$\Longrightarrow 2^9(2^{x-9}-1)=3(3^{y-1}-1)$$ $$\Longrightarrow (x,~y)=\boxed{(9,~1)}$$ İs there any solution?
  9. chwala

    Solve the equation ##x^\frac{17}{6} + x^\frac{21}{25} =15##

    Would we also use Newton's method, or are there more powerful methods? Are there general methods for solving equations with fractional exponents? The solution to this problem is; https://www.wolframalpha.com/input?i=solve+x%5E%2817%2F6%29+%2B+x%5E%2821%2F25%29+%3D+15
  10. chwala

    Solve the given equation that involves fractional exponent

    Is there a better way of solving this, My steps, ##5(x+1)^{1.5}-4x-28=0## ##(x+1)^{1.5}=\dfrac{4x+28}{5}## ##(x+1)^3=(\dfrac{4x+28}{5})^2## ##x^3+3x^2+3x+1=\dfrac{16(x+7)^2}{25}## ##25x^3+75x^2+75x+25=16(x^2+14x+49)## ##25x^3+59x^2-149x-759=0## Letting ##f(x)=25x^3+59x^2-149x-759=0## and using...
  11. L

    Find the equation of g(x)

    I’ve found that when I use g(x)= (x+3)(x-1)2 You get everything except the maximum x=-1 I just cannot get that last one
  12. A

    Find Relativistic Momentum Equation for a Moving Reference Frame?

    I figured since ## dr/dt ## is simply the velocity of the target mass, the velocity ##u_x## would simply have to be changed by the Lorentz transformation. Since the rest mass doesn't change, I think this should be as simple as taking the Lorentz transformation for velocity, and substituting the...
  13. murshiddreamengineer

    I How can we prove the kinetic energy equation

    Proof of kinetic energy work done equals a change in kinetic energy in a mechanical system. δW = F.ds W = ∫F.ds W = m∫a.ds W = m∫(dv/dt).ds W = m∫v.dv here if v and dv are in the same direction the change in kinetic energy will be the usual equation. what happens if both are in different...
  14. A

    I To What Extent Can You Manipulate Limits?

    I'm currently taking Math Methods in Physics, and we're working on infinite series right now. In one of the examples, in order to find the limit, they do what you can see below in "Example 3" As you can see, they turn the original equation into a ln equation, so that they can bring the...
  15. D

    Mathematica Solve equation perturbatively

    I have this expression, $$T=2 P r-\frac{q^2}{4 \pi r^3}+\frac{1}{4 \pi r}$$. Now I want to solve this equation for ##r## perturbatively. This will give the expression $$r=\frac{T}{2 P}-\frac{1}{4 \pi T}+\frac{P \left(8 \pi P q^2-1\right)}{8 \left(\pi ^2 T^3\right)}+.......$$. I was reading...
  16. R

    Energy conservation equation to find equation for final velocity

    I initially thought about the different forms of energy present at each of the points: Total energy at starting point: PEA+ KEA= mgH at point D: KE_D = 1/2mv2f PED= mgD Energy at point D: PED+ KED D = mgD + 1/2 mv2f because EA= ED mgH = mgD = 1/2 mv2f mg(H-D) = 1/2 mv2f g(H-D) = 1/2...
  17. M

    How to prove this without using Cardano's formula?

    Proof: Let ## x=\sqrt[3]{18+\sqrt{325}}+\sqrt[3]{18-\sqrt{325}} ##. Then ## x^3=(\sqrt[3]{18+\sqrt{325}}+\sqrt[3]{18-\sqrt{325}})^3 ##. Note that ## (a+b)^3=a^3+3ab(a+b)+b^3 ## where ## a=\sqrt[3]{18+\sqrt{325}} ## and ## b=\sqrt[3]{18-\sqrt{325}} ##. This gives ##...
  18. S

    Vector parametric equation of line

    I can imagine x + y = 1 to be line in xy - plane but how can x + 2y + z = 3 be a line, not a plane? Thanks
  19. ShivamM

    A Where Can I Find Detailed Derivation of Teleparallel Gravity Field Equations?

    Where can I find detailed derivation of Field Equation of Teleparallel Gravity from variation of Action ?
  20. Heisenberg7

    B Are Implication and Equivalence Interchangeable in Logic?

    I've seen a lot of people use implication and equivalence logic incorrectly. For example, when solving equations (i.e. ##x - 2 = 3 \implies x = 5##). Implication is not reversible, thus it only works in one way. By saying, ##x - 2 = 3 \implies x = 5##, you are essentially saying that it is...
  21. T

    How to balance an equation for the incomplete combustion of acetic acid?

    Balancing the complete combustion of acetic acid equation to carbon dioxide and water is straightforward if you remember hydrogen is always +1, elemental oxygen is zero and combined oxygen is always -2. just balance the exchange of electrons between oxygen and carbon. But how do you balance...
  22. U

    I What year were Navier-Stokes equations introduced?

    Who and when first time introduced below equations(dont have to be in same notation, content is important)? If this formula is always the same, what is contribution of Navier, what of Stokes, what changes all these years?
  23. M

    Euler- Lagrange equation proof

    For this problem, The solution is, However, I have a question about the solution. Does someone please know why they write out ##\frac{dF}{dx} = \frac{\partial F}{\partial y}y' + \frac{\partial F}{\partial y'}y''## since we already know that ##\frac{dF}{dx} = 0##? Thanks!
  24. S

    The value of (b - c) / (c - a)

    $$(b-a)^2-4(b-c)(c-a)=0$$ $$b^2-2ab+a^2=4(bc-ab-c^2+ac)$$ $$b^2-2ab+a^2+4ab=4bc-4c^2+4ac$$ $$(b+a)^2-4ac=4c(b-c)$$ $$b-c=\frac{(b+a)^2-4ac}{4c}$$ I don't know how to continue and not even sure what I did is useful. Thanks
  25. M

    I Separation of variables and the chain rule

    Hi; given the equation ydy/dx=x^2 how is the chain rule applied to result in ydy =x^2dx? Thanks
  26. P

    I Help understanding Maxwell's Equations please

    I have having trouble understanding Maxwell's Equations. Can anyone recommend some good book or website that can help me to understand these Equations? How can electric and magnetic fields travel perpendicular to each other? What causes electromagnetic waves to first radiate from its source? I...
  27. putongren

    Two Trains and a Bee: Distance Question

    This is a question from the MIT Open courseware website. (1). d = vt + ut let t = time it takes d = (u + v)t t = d / (u + v) (2). d = vt + ut d - vt = ut. Substitute t with d / (u + v) d - v*(d/(u+v)) = u*(d/(u+v)) d - v*(d/(u+v)) = “distance...
  28. putongren

    Bees and Trains: A distance problem

    This question is from the MIT Courseware. I’m having difficulty finding the general equation to solve the problem (1). d = vt + ut d = (u + v)t t = d/(u + v) (2). d = vt + ut d - vt = ut sub t with d/(u+v) d - (v*d)/(u+v) = (u*d)/(u+v) I’m done with the...
  29. chwala

    Solve the given trigonometry equation

    I was able to solve with a rather longer way; there could be a more straightforward approach; My steps are along these lines; ##\sinh^{-1} x = 2 \ln (2+ \sqrt{3})## ##\sinh^{-1} x = \ln (7+ 4\sqrt{3})## ##x = \sinh[ \ln (7+ 4\sqrt{3})]## ##x = \dfrac {e^{\ln (7+ 4 \sqrt{3})} - e^{-[\ln 7+ 4...
  30. D

    Mathematica Solving a complicated equation for approximate analytical Solution using Mathematica

    Hello there, I am trying to solve the Following equation for r, $$2 a Q^4+5 r^4 \left(3 c (\omega +1) r^{1-3 \omega }-2 r (r-3 M)-4 Q^2\right)=0$$ Clearly this is unsolvable. But if we substitute a=0 and c=0 we get one of the solution, ##r=\frac{1}{2} \left(\sqrt{9 M^2-8 Q^2}+3 M\right)##. Can I...
  31. atky1224

    Simultaneous equation involving cos, sin

    Thanks a lot in advance!
  32. chwala

    Solve the given first order PDE

    Solve the given PDE for ##u(x,t)##; ##\dfrac{∂u}{∂t} +10 \dfrac{∂u}{∂x} + 9u = 0## ##u(x,0)= e^{-x}## ##-∞ <x<∞ , t>0## In my lines i have, ##x_t = 10## ##x(t) = 10t+a## ##a = x(t) - 10t## also, ##u(x(t),t)= u(x(0),0)e^{-9t}## note this is from, integrating ##u_t[u(x(t),t] =...
  33. chwala

    Solve the given first order Partial differential equation.

    Solve the given PDE for ##u(x,t)##; ##\dfrac{∂u}{∂t} +8 \dfrac{∂u}{∂x} = 0## ##u(x,0)= \sin x## ##-∞ <x<∞ , t>0## In my working (using the method of characteristics) i have, ##x_t =8## ##x(t) = 8t + a## ##a = x(t) - 8t## being the first characteristic. For the second...
  34. S

    Confusion about variables in polar coordinates

    My confusion refers to this question above. If I were to ask you, what is the equation of the radial line, what would you say? I know that the general equation the radial line with cartesian gradient of m has an equation of θ = arctan(m). Clearly here the angle between the radial line and...
  35. S

    How do I use LaTex in the forum for to write equations?

    Hello. I registered today. Maybe you can help me. How do I use LaTex in the forum for to write equations?
  36. T

    Help with Recurrence Equation

    Hi there, I am going through a book on multi-storey steel structures and I have come to a chapter that gives approximate methods to calculate rotations at the joints (The intersecting members) of a rigid frame. There is a recurrence equation that computes the rotations and this is given below...
  37. V

    I Normal Mode calculation steps

    Can someone please explain me the steps of calculation of X1:X2 after putting in the lower value of W^2 in equation 9.9 in "Riley, Hobson, Bence - Mathematical Methods for Physics and Engineering 2006 - pg 319"? I have attached the page as a PDF file. Thank you.
  38. D

    B An exact Paraxial equation derivation, 100% Cartesian

  39. M

    Solving linear DE systems using fundamental matrix

    For this problem, I am confused by the term below. I get all their terms, expect replacing the highlighted term by ##e^{3t}##, does someone please know whether this is yet another typo? Thanks!
  40. D

    B Questions about the Paraxial region for a single spherical surface

  41. paulb203

    Is this a simultaneous equation question?

    hb=54 2h+2b=33 h=54/b therefore, 2(54/b)+b=33 108/b + b = 33 I’ve got a feeling I’ve gone down a blind alley here. Any hints?
  42. chwala

    Find the equation of the tangent plane and normal to a surface

    In my line i have, ##\dfrac{∂r}{du} = \vec{i} +\dfrac{1}{2}u \vec{k} = \vec{i} +1.5 \vec{k}## ##\dfrac{∂r}{dv} = \vec{j} -\dfrac{1}{2}v \vec{k} = \vec{j} -0.5\vec{k}## The normal to plane is given by, ##\dfrac{∂r}{du}× \dfrac{∂r}{dv} = -\dfrac{3}{2} \vec{ i} + \dfrac{1}{2}\vec{j}+\vec{k}##...
  43. ZeroX4

    Looking for equation/formula for calculating boomerang trajectory based on few variables for game

    So basically i am vary green when it comes to equations/formula Well to the point i don't even know if proper name for what i'm looking for is called equation or formula but let's stick to equation So going as simple as i can I want to throw boomerang creating boomerang like ellipse trajectory...
  44. tellmesomething

    Electric field vector equation: Finding the neutral point for two charges

    This is the general suggested approach given in a textbook. My question is why can I not directly write it in vector form? E1 vector + E2 vector =0 should be valid no? Why are they choosing to write E1 mag + E2 mag=0 Then find a vector form Then convert the magnitude equation into a vector...
  45. M

    Lagrange equation for block and incline

    For this problem, Does someone please know where the term highlighted in blue came from? Thanks!
  46. jojosg

    Chemistry Need help with Ideal Gas Question

    Need help solving this question. Can't seem to get the right answer using PV/T=constant P1V1/T1 = P2V2/T2 Patm = 75.23cmHg T1+20+273=293K STP: P=1.01 x 10^5 N/m^2 Pabs=41cmOil P1 = density x g x h = (810 kg/m^3)(9.8 m/s^2)(75.23-41)x10^-2 mOil=2717.18 N/m^2...
  47. 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}...
  48. 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...
  49. 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##
  50. 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}}##...
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