solution Definition and 27 Threads

My solution: Let ##f(x+y)=f(x)f(y)-[f(x)+f(y)]-1/2## İf ##y=x## we find functional equation that given us. So for ##x=y=1## then ##f(2)=-1/2## İf we evaluate ##x=1, y=2## at above equation ##f(3)=-3## My question is: What is the solution of that functional equation; I mean are there other...

From the original equation of ## \ddot{x}+x+\frac{3}{2}\beta x\lvert x\rvert=0 ##, I've got ## \ddot{x}\dot{x}+x\dot{x}+\frac{3}{2}\beta x\lvert x\rvert\dot{x}=0 \implies \frac{d}{dt}(\frac{1}{2}\dot{x}^2+\frac{1}{2}x^2+\frac{\beta}{2}x^2\lvert x\rvert)=0 \implies...

So far, I've got that ## g(a)=(c-1)a+\frac{3}{4}a^3\implies g'(a)=c-1+\frac{9}{4}a^2 ##. I know that if the first derivative of a function is positive (greater than ## 0 ##), then that function is always/strictly increasing. However, how should I construct this proof in order to show that...

In my case, there is proton radiation acting on the material. Consequently, the protons get stuck in the sample and create an electrostatic field. I would like to solve the Poisson equation inside the sample. I assume that the medium is infinite and homogeneous, that is, the potential at...

In the part below where the author of this solution, wants to show that: ##a\vee (b\wedge c)=(a\vee b)\wedge (a\vee c)##, in the third equality he adds ##2ac+2ab+2abc=2a\wedge(b\vee c)##; can someone explain to me why can we do this here? Thanks!

Iv'e got this proof from some claim: Any Pure maths doctors out there who can explain to me why since ##V\ne \emptyset## that ##1\notin \mathfrak{b}##? Thanks!

I've never had a good understanding of this. Say I am dissolving 20g sodium chloride into 1L of water. Is the only way to theoretically predict the final volume of solution to consult density tables? edit: I realize this isn't a practical situation, and that typically you are making a solution...

I didn't have any good way to put this in the homework statement, but this is what the question is asking: For what c-value(s) will there be a non-trivial solution: ## x_1 - x_2 + x_3 = 0 ## ## 2x_1 + x_2 + x_3 = 0 ## ##-x_1 + (c)x_2 + 2x_3 = 0 ## I have spent a good couple hours looking at...

TL;DR Summary: In Morin 3.7 sliding sideways on a plane I used a completely different method than he did and got the correct answer is my method right The problem statement is as follows I split up the friction force into x and y components derived a diff eq for v_y in terms of v_x then took...

So I tried it out by taking a patch of area da at an angle theta from the x axis and rotating it around the axis, this gives you a cone whose, locus is that of a uniformly charged ring since all of the area is at the same angle theta and the surface charge density varies with theta. My solution...

For this problem, My working is, ##0v_1 + 0v_2 = 0##, however, does someone please know why the example says we cannot choose ##v_1 = (0, 0)## since from ##0v_1 + 0v_2 = 0## ##v_1, v_2 \in \mathbb{R}## i.e there is no restriction on what the vector components could be)? Thanks!

My interest is on the highlighted (In Red). Otherwise the other steps are clear. We have on that part of the problem, ##(-p\sin t -q\cos t)-12(p\cos t -q \sin t)+36p\sin t +36q\cos t = 37 \sin t + 0 \cos t## Ah I just realized we are solving a simultaneous equation for ##p## and ##q## ...

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##

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...

Can someone please show me the steps from circled equation 1 to 2? Thank you.

Hi friends, I was wondering if you could give the definition of 'closed form', with examples of closed form solutions and open? form solutions. Foe example, is this a closed form solution? $$\sum_{k=1}^\infty \frac{1}{2^k}$$ Or this? $$\sum_{k=1}^5 \frac{1}{2^k}$$ Thanks.

I have never solved a matrix ODE before, and am wondering if solving it is similar to solving ##y'=ay## where ##a## is a constant and ##y:\mathbb{R} \longrightarrow \mathbb{R}## is a function. The solution is right according to wikipedia, and I am just looking for your inputs. Thanks...

I am currently engaged in a local project that requires a unique attachment solution. We need to securely affix a 100g object, coated with cross-linked polyethylene foam, to a board that is covered in synthetic fur. The attachment must be robust enough to endure shaking and bumps. Importantly...

I cannot find much on the Lentz solution for superluminal travel since the initial reactions to the paper coming out in 2021. Has more mathematical work been done on this? I have my doubts it would work based on some of the criticisms I have seen but want to see if someone has really evaluated...

This is my solution and it remained still and did not bubble once while the pot boiled . I realise salt and oil impurities may increase the boiling point but I wouldn't imagine to this extent, and when I filtered the solution the oil seemed to collect the copper and other dissolved metals...

Looks pretty straightforward, i approached it as follows, ##f_x = u(x+cy) + v(x-cy)## ##f_{xx}=u(x+cy) + v(x-cy)## ##f_y= cu(x+cy) -cv(x-cy)## ##f_{yy}=c^2u(x+cy)+c^2v(x-cy)## Therefore, ##f_{xx} -\dfrac{1}{c^2} f_{yy} = u(x+cy) + v(x-cy) - \dfrac{1}{c^2}⋅ c^2 \left[u(x+cy)+v(x-cy)...

During calculating null space from rref matrix some rows are with 1 variable so it give this variable value of zero so the null space of rank deficient matrix be for example {-1,2,0,3,-4,0} my question is how to get rid of zeros in the null space solution and only solve it as basic and free...

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...

This has bothered me for a while and I have not found a good solution, it's convenient to write math equations using the \$\$ on the fly and occasionally we want to put a tag for an equation. tag works for the equation environment but not for \$\$. For example I would like to have [1] displayed...

Hi, PF 1-The elementary integral is ##\displaystyle\int{\displaystyle\frac{1}{a^2+x^2}dx}=\displaystyle\frac{1}{a}\tan^{-1}\displaystyle\frac{x}{a}+C## 2-The example is...

Hi, unfortunately, I have a problem to solve the following task The equation looks like this: $$\left(\begin{array}{c} \frac{d}{dt} x(t) \\ \frac{d}{dt} y(t) \end{array}\right)=\left(\begin{array}{c} -a y(t) \\ x(t) \end{array}\right)$$ Since the following is true ##\frac{d}{dt}...

Hardly anyone would think of this solution. I even wonder how anyone would come up with the question. And it is even more amazing that the questioner and solver were not the same person.