Separable Definition and 185 Threads

From Griffiths 3rd edition quantum pg 45 The bottom two lines. Why can the general solution be expressed as that sum of separable solutions? Griffith doesn't seem to explain this step. What am I missing? Thank you.

In a central potential problem we have for the Hamiltonian the expression: ##H=\frac{p^2}{2m}+V(r)## and we use to solve problems like this noting that the Hamiltonian is separable, by separable I mean that we can express the Hamiltonian as the sum of multiple parts each one commuting with the...

This is a physics problem from Griffith's Electrodynamics. I'm mainly asking about the math here. I found the DE in the box at part (d). To solve it, I did: ##\sqrt V {d^2 V} = \beta dx^2## Integrating twice: ##\frac {4} {15} V^{2.5} = \beta x^2/2## Why is my method wrong? Thanks for the help.

Greetings, I have a question to the following section of the book https://www.springer.com/gp/book/9783319163741: I understand that the equation is separable, since I can just write $$ \int_{x_0}^{x} \frac {1}{V(x', \xi, \eta)}dx' =\int_{0}^{t}dt' .$$ However, without knowing the exact shape...

I fell upon such an equation : $$-E'(v)a(1+\frac{cE(v)}{\sqrt{E(v)^2-1}})=vE(v)+c\sqrt{E(v)^2-1}$$ It's not separable in E on one side and v expression on the other. So I'm looking for methods to solve this maybe changes of coordinates ?

I introduced the unitary transformation ##U=U_a \otimes U_b## with ##(U_a\otimes 1):\;|s,s> \rightarrow\,\frac{1}{\sqrt{d}}\sum_t \omega^{ts}|t,s> ## und ##(1\otimes U_b):\;|s,s> \rightarrow\,\frac{1}{\sqrt{d}}\sum_t \omega^{-ts}|s,t> ## ##(\omega=e^{2\pi i/d}##) and let it act on the state in...

1000 use Separable Equations to solve $$y'= \frac{x^2}{y(1+x^3)}$$ Multiply both sides by the denominator $$y(1+x^3)y'=x^2$$ Subtract $x^2$ from both sides $$-x^2 +y(1+x^3)y'=0$$ ok was trying to follow an example but ?

2000 $\textsf{solve the given differential equation}$ $$y'=\frac{x^2}{y}$$ ok this is a new section on separable equations so i barely know anything but wanted to post the first problem hoping to understand what the book said. thanks ahead...

Homework Statement ##(y-1)dx+x(x+1)dy=0## Homework EquationsThe Attempt at a Solution [/B] I multiplied both equation with, ##\frac {1} {(y-1)x(x+1)}## so I get ##\frac {dx} {x(x+1)}+\frac {dy} {y-1}=0## taking integral for both sides then I get ##ln(x)-ln(x+1)+ln(y-1)=ln(c)## so ##ln(\frac...

Hi at all On my math methods book, i came across the following Fredholm integ eq with separable ker: 1) φ(x)-4∫sin^2xφ(t)dt = 2x-pi With integral ends(0,pi/2) I do not know how to proceed, for the solution...

I am reading David S. Dummit and Richard M. Foote : Abstract Algebra ... I am trying to understand the proof of Proposition 37 in Section 13.5 Separable and Inseparable Extensions ...The Proposition 37 and its proof (note that the proof comes before the statement of the Proposition) read as...

Dummit and Foote in Section 13.5 on separable extensions make some remarks about separable polynomials that I do not quite follow. The remarks follow Corollary 34 and its proof ... Corollary 34, its proof and the remarks read as follows: In the above text by D&F, in the remarks after the...

I am reading Paul E Bland's book: The Basics of Abstract Algebra and I am trying to understand his definition of "separable polynomial" and his second example ... Bland defines a separable polynomial as follows: ... and Bland's second example is as follows: I am uncomfortable with, and do...

I am reading both David S. Dummit and Richard M. Foote : Abstract Algebra and Paul E. Bland's book: The Basics of Abstract Algebra ... ... I am trying to understand separable polynomials ... ... but D&F and Bland seem to define them slightly differently and interpret the application of the...

Homework Statement (a) Consider a cylindrical can of gas with radius R and height H rotating about its longitudinal axis. The rotation causes the density of the gas, η, to obey the differential equation dη(ρ)/dp = κ ω2 ρ η(ρ) where ρ is the distance from the longitudinal axis, the constant κ...

Dear forum, I am trying to understand what a separable vector space is. I know we can perform the tensor product of two or more vector space and obtain a new vector space. Is that vector space separable because it is the product of other vector spaces? thanks

Homework Statement \frac{dy}{dx}\:+\:ycosx\:=\:5cosx I get two solutions for y however only one of them is correct according to my online homework (see attempt at solution) Homework Equations y(0) = 7 is initial condition The Attempt at a Solution \int \:\frac{1}{5-y}dy\:=\:\int...

Hey! :o In my notes there is the following: Let $F$ be a field. The irresducible $f\in F[x]$ is separable, if all the roots are different. A non-constant polynomial $f\in K[x]$ is separable, if all the irreducible factors are separable. Example: $f(x)=(x^2-2)^2(x^2+3)\in \mathbb{Q}[x]$...

Hey! :o Let $E/F$ be a finite extension. I want to show that this extension is Galois if and only if $E$ is a splitting field of a separable polynomial of $F[x]$. I have done the folllowing: $\Rightarrow$ : We suppose that $E/F$ is Galois. So, we have that the extension is normal and...

Hi, I am learning ODE and I have some problems that confuse me. In the textbook I am reading, it explains that if we have a separable ODE: ##x'=h(t)g(x(t))## then ##x=k## is the only constant solution iff ##x## is a root of ##g##. Moreover, it says "all other non-constant solutions are separated...

Hey! :o Let $F$ be a field, $D=F[t]$, the polynomial ring of $t$, with coefficients from $F$ and $K=F(t)$ the field of rational functions of $t$. (a) Show that $t\in D$ is a prime element of $D$. (b) Show that the polynomial $x^n-t\in K[x]$ is irreducible. (c) Let $\text{char} F=p$. Show...

Problem: y'=((x-1)/(x^2))*(y^2) , y(1)=1 . Find solutions satisfying the initial condition, and determine the intervals where they exist and where they are unique. Attempt at solution: Let f(x,y)=((x-1)/(x^2))*(y^2), which is continuous near any (x0,y0) provided x0≠0 so a solution with y(x0)=y0...

Homework Statement Solve the differential equation: (ex+1)cosy dy + ex(siny +1)dx=0 y(0)=3 Homework Equations none The Attempt at a Solution (ex+1)cosy dy + ex(siny +1)dx=0 (ex+1)cosy dy =- ex(siny +1)dx cosy/(siny+1)dy=-ex/(ex+1)dx ∫cosy/(siny+1)dy=-∫ex/(ex+1)dx using u sub on both the...

Homework Statement Solve each of the following differential equations: 4xydx + (x2 +1)dy=0Homework Equations None The Attempt at a Solution 4xydx + (x2 +1)dy=0 (x2 +1)dy=-4xydx dy/y=-(4xdx)/(x2 +1) ∫dy/y=∫-(4xdx)/(x2 +1) ln|y|=-2ln|x2+1| +C used u-sub on last step fo u=x2 +1

Solve the separable differential equation $\displaystyle y'=x^4y^4$ Solve for $y$ if possible. $\displaystyle y=\frac{{y'}^{(1/4)}}{x}$ Not sure ?

Homework Statement Q=-1*K(T)*(H*W)*(dT/dx)+((I^2)(p)(dx)/(H*W)) K(T)=(197.29-.06333333(T+273)) H=0.01905 W=0.06604 I=700 p=10*10^-6 Q=some constant Please separate and differentiate to solve for Q using variables of T and x. Boundaries: T: Upper=T1 (constant) Lower=T0 (constant) x: Upper=L...

I am really struggling with this one, if anyone can help. (ln(y))3*(dy/dx)=(x^3)y with initial conditions y=e^2 x=1 I get c=4/(e^4) - 1/4 then I get stuck at (3ln^2y - ln^3y)/(y^2)=(x^4)/4 + C Any ideas? I'm really not good at these so there are probably mistakes, because at this point I have...

4 du/dt = u^2 with initial condition u(0)=6 I have worked this multiple times, and all I get is u = (-8/(t-27))^(1/3) and it is NOT right! If anyone can help it would be very appreciated.

Homework Statement i am asked to form a differential equation using dy/dx = 1 + y + (x^2 ) + y(x^2) , but i gt stucked here , homework to proceed? as we can see , the V and x are not separable Homework EquationsThe Attempt at a Solution

So It is well known that the 2D solution to the Laplace equation can be obtained by changing to complex coordinates ##u=x+iy## and ##v=x-iy##. This can be extended to n dimensions as long as the complex coordinates chosen also solve the Laplace equation. For example in 3D...

Let ##\mathbb{X}## be the set of all sequences in ##\mathbb{R}## that converge to ##0##. For any sequences ##\{x_n\},\{y_n\}\in\mathbb{X}##, define the metric ##d(\{x_n\},\{y_n\})=\sup_{n}{|x_n−y_n|}##. Show the metric space ##(\mathbb{X},d)## is separable. I understand that I perhaps need to...

do you think of Bell's theorem when you think of forming a couple : that it builds a unity and is not separable ?

Hi. Bell's formulation of local realism is $$P(a,b)=\int\ d\lambda\cdot\rho(\lambda)p_A(a,\lambda)p_B(b,\lambda)\enspace.$$ Let's for simplicity assume there's only a finite number of states, so this becomes $$P(a,b)=\sum_{i} p_i\cdot\ p_A(a,i)p_B(b,i)\enspace.$$ I'm trying to translate this...

Homework Statement It is just an evaluation problem which looks like this dx/dy = x^2 y^2 / 1+x Homework Equations dx/dy = x^2 y^2 / 1 + x The Attempt at a Solution What i did is cross multiply to get this equation y^2 dy = x^2 / 1+x dx then next line ∫y^2 dy = ∫x^2/1+x dx y^3/3 = ∫dx + ∫1/x...

Homework Statement What is the general solution of: y'=(3*y^2-x^2)/(2*x-y) Homework EquationsThe Attempt at a Solution This First Order equation is neither linear nor separable. I also have checked the Exact test, which turns to be Not Exact. Any help regarding how...

Homework Statement I believe I have solved this differential equation, yet do not know how the book arrived at it's answer... Solve the differential equation in its explicit solution form. The answer the book gives is... Homework Equations Separable Differential Equation The Attempt...

Homework Statement Solve the differential equation, explicitly. dy/dx = (2x)/(1+2y) The answer given by the book is... -1/2 + 1/2sqrt(2x - 2x^2 +4) Homework Equations Process for solving separable differential equations The Attempt at a Solution dy/dx = (2x)/(1+2y) (1 + 2y)*dy = 2x*dx...

Homework Statement Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable. (Rudin: Principles of Mathematical Analysis, 2nd ed.) Homework Equations Every separable metric space has a countable base. The...

Homework Statement [/B]Homework Equations The Attempt at a Solution I've highlighted two equations on the screenshot. How did it proceed from the first to the second? I'm actually confused with the absolute values. What is the idea behind getting rid of the first absolute value(1-5v^2) while...

Hi everyone, I am trying to find any particular solution for the equation dy/dx + y = 1. I have been told it is not separable. I have done the following: dy/dx = 1-y integral of 1/(1-y) dy = integral -loge(1-y) = c e^-c = 1-y y = 1- e^-c let c = 0 y = 1-1 A particular solution is y= 0. My...

Homework Statement I have two equations. cos(θ)wφ + sin(θ)wφ = 0 (1) And ## \frac{w_φ}{r}## + ∂wφ/∂r = 0 (2) Find wφ, which is a function of both r and theta. Homework EquationsThe Attempt at a Solution I end up with two equations, having integrated. wφ=## \frac{A}{sinθ}## from (1)...

When solving a separable differential equation, my textbook says this: ln|v-49|=-t/5+C→ |v-49|=e-t/5+C→ v=49+ce-t/5 What happened to the absolute values? I think it has something to do with the exponential always being positive.

What is getting separated from what? I presume there is some historical founding case that involved separating something. Like how the original vector spaces were mental arrows in R^3.

Find the solution of the given initial value problem in explicit form. Determine interval which solution is defined. (which i think is the same thing as saying find the interval of validity) $y' = (1-2x)y^2$ , $y(0) = -1/6$ So here is what I have so far.. $\int y^{-2}dy = x - x^2 + C$ $=...

(a) Find the solution of the given initial value problem in explicit form. (b) Plot the graph of the solution (c) Determine (at least approximately) the interval in which the solution is defined $$\frac{dr}{dx} =\frac{r^2}{x} $$ and $$ r(1) = 2$$ I'm kind of confused..How do I start this problem?

I understand how to integrate this: ∫y2dy. I don't understand how to integrate this: di(t)/dt = i(t)p(t) intergrate((di(t)/dt/i(t))*dt = p(t)dt) (see this image: http://i.imgur.com/OdKI309.png) how do you perform the intergral on the left, seeing as as it not dt, but di(t)? thanks

Homework Statement 2y * y'(t) = 3t^2 such that y(0) = 9 Homework Equations g(y)y'(t) = h(t) The Attempt at a Solution So I have done many of these seperable ones in homework that did not require a parameter so now I got lost. This is what I did;[/B] Integral of 2y dy = integral of 3t^2...

Homework Statement dx/dt=x^2+1/25, and find the particular solution satisfying the initial condition x(0)=8. Homework EquationsThe Attempt at a Solution So I began by taking out 1/25 from the right side, making the equation: dx/dt = (1/25)(25x^2 + 1) Then, rearranging the equation to be...

Hello. I need some help solving a differential equation. I think where I'm going wrong is integrating one side via partial fractions, but I'm not quite sure where my mistake is. Using Wolfram, I found the correct answer, which is below. Thanks. Homework Statement Solve the following initial...

I have this equation: dy/dx = 1-y^2, so then dy/(1-y^2) = dx, so ∫dy/(1-y^2) = dx ---> ∫(A/(1+y) + B/(1-y))dy = x + C. I rewrite it again: (A - Ay + B + By)/(1+y)(1-y) = 1/(1+y)(1-y) so I get A+B = 1, and B-A = 0, so B = A, and therefore 2A = 1. so A & B = 1/2. So 1/2∫(dy/(1+y) +...