What is Analytic solution: Definition and 30 Discussions
In mathematics, a closed-form expression is a mathematical expression expressed using a finite number of standard operations. It may contain constants, variables, certain "well-known" operations (e.g., + − × ÷), and functions (e.g., nth root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit, differentiation, or integration. The set of operations and functions admitted in a closed-form expression may vary with author and context.
Is there an analytic solution to the simple equation ##x^{\alpha} +x =1##? I can get to a solution by iteration or by graphical methods but I wish to find a closed form exact solution. ##\alpha## is a constant. I tried to put it into a form where I could use the quadratic formula but that didn't...
Three (many) body problems where three or many bodies (particles) interact are impossible to solve analytically. First one appeared in classical mechanics where equations of motion of planets were tried to be found by applying Newton's 2nd law for system of planets and stars interacting via...
At the point where we 'guess' a solution to this 2nd order ODE that cannot be done analytically, I was wondering why Griff and others choose $$e^{-x^2 / 2}$$ rather than just $$e^{-x^2}$$ I've plotted both here and am left wondering what's so different? If we guessed instead the unpopular...
Hi,
I think I'm having a bit of a brain fart...I'm messing with this numerical code trying to understand the 1-D time-independent Schrodinger's equation infinite square well problem (V(x) infinite at the boundaries, 0 everywhere else). If normalized Phi squared is the probability of finding...
The exercise is to compare numerical and analytical solution. I have worked out the code from earlier exercise (see code under this text), but I don't understand how the analytical solution works. I have tried to use the equation r(theta) = a(1-e^2)/(1+e*cos(theta)), which is OK but I don't...
I am trying to write code for analytical solution of 1D heat conduction equation in semi-infinite rod. The analytical solution is given by Carslaw and Jaeger 1959 (p305) as
$$
h(x,t) = \Delta H .erfc( \frac{x}{2 \sqrt[]{vt} } )
$$
where x is distance, v is diffusivity (material property) and t...
Nowadays much of the community of astronomy and astrophysics is focused on simulations and data analysis. Thus people, who do not have access to grids and data, must resort to analytical analysis. What are the challenging areas of astronomy and astrophysics that rely on analytical workout rather...
Homework Statement
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Plot the solution at times t = 0, t = 0.25 and t = 0.5. Based on your knowledge of the general solution of the wave equation, the behaviour you see here, and the initial data profile, what is the analytic solution u(x, t) to this problem? (In deriving the analytic...
Hello all, first post.
I have come here to get second opinions on the program I have written to compute the Einstein Tensor (the Riemann Tensor and Ricci Tensor). I enjoy looking for solutions to the Einstein Field Equations, however computing them by hand is not realistic. I decided to write a...
The quantum harmonic oscillator is an analytic solution of the Schrodinger Equation. Does the original Dirac Equation for a free electron also have an analytic solution? Of course a "solution" of the Dirac Equation would consist of 4 functions.
Thanks in advance.
I have what I think is a valid solution, but I'm not sure, and when I try to check the answer approximately in Matlab, I don't get a verified value, and I'm not sure if my analytic solution or my approximation method in Matlab is at fault.
1. Homework Statement
Evaluate the integral...
I have the following system of equations with variables ##a,m##, and I'm wondering—can this system be solved symbolically/analytically?
\begin{align}
m &= 100 + \frac{ \left( 200 \frac{\ln{\frac{1}{2}}}{26.8} \right) }{\left(\dfrac{\ln{\frac{1}{2}}}{26.8} + a \right)}
\\ \\
50 &= me^{-a\left(...
Homework Statement
The question is in the attached image . My problem starts when dealing with the limits of integration . I need an analytic procedure of solving such problems without involving graphical method . The equations of the graphs of h(t) and x(t) are easily derived .
Homework...
Homework Statement
the function is Exp[-w^2]/w^2, how to solve the Fourier transform analytically with Residue theorem?
It is better if there is more general results.
Mathematica can solve it analytically, but I need a human-soluable way.
Homework Equations
The Attempt at a...
Falling polygons: meshing vs. stacking -- analytic solution needed
I'm a game developer and not a mathematics specialist, so I'm not 100% sure if this question is correctly categorized.
My problem is as follows.
I'm building a game that's similar to Tetris, but in 3D instead of 2D...
I have been trying to find an analytic solution to the time-dependent Schrodinger Equation. I plan to make a movie of the probability function as it changes over time, but I can't seem to find any analytic solution for the wave function.
Is it possible to solve the time-dependent Schrodinger...
I have the PDE:
(v_r)^2+(v_z)^2=p^2 where v=v(r,z), p=p(r,z).
I have some boundary conditions, of sorts:
p=c*r*exp(r/a)exp(z/b) for some constants a,b,c, at r=infinity and z=infinity
p=0 at f=r, where
(f_r)^2=p*r/v-v*v_r
(f_z)^2=p*r/v+v*v_r
Is it possible that one could obtain an...
Is there a general analytic solution to the classical motion of a relativistic charged particle in a static Coulomb potential? Of course, the non-relativistic limit is simply Kepler's problem. Quantum effects should be ignored, but relativistic effects (such as E field transforming into B field)...
Homework Statement
Consider the following ODE
x^3y' - 2y + 2x = 0
Homework Equations
Prove that the ODE has no analytic solution in any neighborhood of x=0 .
The Attempt at a Solution
Its general solution is Ce^{-\frac{1}{x^2}}+ e^{-\\frac{1}{x^2}} \int_{x_0}^x -...
Hi,
I have a question about integrating an numeric solution to a differential equation in Maple. I have solved a system of 3 odes:
sol := dsolve({initial, syst}, func, numeric)
where
syst := diff(OmegaLambda(x), x) = ode1, diff(OmegaK(x), x) = ode2, diff(lnH(x), x) = ode3.
This then gives me...
Homework Statement
I wonder if someone could help me to arrive at equation 2.56 by performing the substitutions. Please see the attachment
Homework Equations
Please see the attachment for this part. and also for the attempt of a solution.
Hello,
I am trying to find an analytic solution to the following:
\int_{-1}^{1}\exp(-p\sqrt{1-x^{2}}-qx)dx
where p,q > 0.
Does anyone have any ideas? Thanks.
Homework Statement
y'=4e^{0.8x}-0.5y
This question was obtained from a textbook, where it is used as an example for the application of Heun's method (of ODE integration). They state that it has a 'simple' analytic solution of,
y=\dfrac{4}{1.3}(e^{0.8x}-e^{-0.5x})+2e^{-0.5x}
Homework...
Homework Statement
Does the following differential equation have a simple analytic solution?
\frac{dy}{dx}=y-\frac{2x}{y}
Homework Equations
Don't know.
The Attempt at a Solution
I would have said that there is no simple solution to this equation, I tried to find the integrating factor...
Hi everyone. Maple and I have collectively racked our brains and I've tried most of the integration techniques I know. Does anyone know the solution to the integral
\int exp(-a*abs(x))*exp(i*(k0-k)*x)*dx
from -infinity to infinity (not sure how to get the limits over the integral sign...
(not a HW problem- I'm writing a paper)
Looking for an analytic solution for the following- if one exists
\frac{1}{2\pi}\int_0^{2\pi}e^{-cos^2(\theta)}d\theta
I recently made an interesting problem involving a uniform rod. I would like to find an analytic solution to this problem however I do not know if this is possible. The equation yielding the equilibrium points is given by
omega^2*R*cos(phi) + omega^2*(L/2)*cos(phi)*sin(phi) + g*sin(phi) = 0...
I am looking for the analytic solution of this ODE (if it were one):
s^2G''+sG'-(1+s^2+s\text{coth}(s))G=-4s^2e^{-s}
I have solved this equation numerically, it only gives one physically realizable configuration rejecting conveniently one of the homogenous solutions. I don't have those...