equation Definition and 125 Threads

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...

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...

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...

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...

Thanks a lot in advance!

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] =...

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...

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...

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

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...

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.

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!

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?

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}##...

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...

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...

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

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...

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}...

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...

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##

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}}##...

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)##...

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...

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...

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...

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

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...

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##...

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...

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...

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 =...

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.

I've attached an example here.

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?