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.
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: $$P^2\frac{\partial^2P}{\partial x\partial y}=(P^2+1)\frac{\partial P}{\partial x}\frac{\partial P}{\partial...
Homework Statement: What actually is the particular solution of an ODE?
Relevant Equations: x
Consider the differential equation ##y'' + 9y = 1/2 cos(3x)##, if we wish to solve this we should first solve the auxiliary equation ##m^2 + 9 = 0## giving us ##m=3i,-3i##, this corresponds to the...
Hello, I need to solve the commutator relations above. I found the equation above for the last one, but I am not sure, if something similar applys to the first one. I am a little bit confused, because I know there has to be a trick and you don't solve it like other commutator.
Thanks for your help!
Hi all,
I am starting with the following equation: ##2\cot\left(\frac{\theta}{2}\right) = \cot\left(\frac{k_{1}}{2}\right) - \cot\left(\frac{k_{2}}{2}\right)##
with the following definitions: ##k_{1} = \frac{K}{2} + ik, k_{2} = \frac{K}{2}-ik, \theta = \pi(I_{2}-I_{1}) + iNk##, where...
Let X be a continuous-time Markov chain that hops between two states ##\{1, 2\}## with rates ##\lambda, \mu>0##, so its generator is
$$Q = \begin{pmatrix}
-\mu & \mu\\
\lambda & -\lambda
\end{pmatrix}.$$
Solve ##\pi Q = 0## for the stationary distribution, and verify that...
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...
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...
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...
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.
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
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...
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...
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...
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!} =...
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...
## \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##.
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...
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?
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}...
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.
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...
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)## ?
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##
...
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...
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...
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...
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...
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...
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...
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...
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...
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...
What is the difference between these two concepts? An equation is said to be "invariant" under some operation if the form of the equation doesn't change. However, isn't that exactly what "covariance" in physical laws means—that the form of the laws remains unchanged when applying an operation to...
Hi.
If I write any random equation in 2D then the graph undoubtedly shows up on that 2D graphing system.
Equation example: ##x^2 y^2 + x^2 y + x y =1##
My question is: if I take the same equation: ##x^2 y^2 + x^2 y + x y =1## and if I manipulate the equation by including another variable...
Hi.
I don´t know if this question should be in the maths forum, but as it´s related with circuit analysis, I will post it here. I just would like to know how you get:
v(t) = 1/C ∫tt0 i(τ) dτ + v(t0)
From:
v(t)=1/C ∫t-∞ i(τ) dτ
I just know the basics of calculus and I don´t know how to...
I am confused with the equation to be used for capacitor in electrical analysis
The standard equation we have is Q=CV -> 1
the other equation is is V = Z*I ohms law Z is the impedance of the capacitor. Both are giving me voltage, which one to use ?
I encountered a problem in reading Phys.Lett.B Vol.755, 367-370 (2016).
I cannot derive Eq.(7), the following snapshot is the paper and my oen derivation,
I cannot repeat Eq.(7) in the paper.
##g^{\mu\nu}## is diagonal metric tensor and##g^{\mu\mu}## is the function of ##\mu## only...