http://www.physicsforums.com Solving ODE and PDE en Mon, 20 May 2013 08:20:50 GMT vBulletin 60 http://www.physicsforums.com/images/physicsforums/misc/rss.jpg http://www.physicsforums.com http://www.physicsforums.com/showthread.php?t=692546&goto=newpost Sun, 19 May 2013 18:21:58 GMT Hi there, my version of Mathematica may be too old and i'm not finding this one by hand so any help would be appreciated:
ψ''(z)=[k²/4 –M² –kδ(z)]ψ(z),
where δ(z) is the Dirac delta, k and M constants.

i can solve the same equation without the M^2 term by exp(k|z|/2), but this one proves to be much more complicated.

Please note that this is part of a problem where solving the D.E. is actually not the issue: i'm supposed to use a software or anything that helps so i'm really just trying to get the answer.. thanks!! ]]> Differential Equations Goddar http://www.physicsforums.com/showthread.php?t=692546 http://www.physicsforums.com/showthread.php?t=692470&goto=newpost Sun, 19 May 2013 08:32:52 GMT Hi Everyone,

I was studying coordinate transformation and I came across this equation, that I couldn't understand how it came up. Let me put it this way:
x = rcosθ

Then if I want to express the partial derivative (of any thing) with respect to x, what would be the expression? i.e. ∂/∂x=?
According to the text, answer would be
∂/∂x=cosθ*∂/∂r - (sinθ/r)*∂/∂θ

Please explain to me how to come up with this expression.

Thnaks ]]> Differential Equations onchoa http://www.physicsforums.com/showthread.php?t=692470 http://www.physicsforums.com/showthread.php?t=692420&goto=newpost Sat, 18 May 2013 23:32:38 GMT Hello everyone,

I need some help on doing convergence tests (comparisons I believe) on some Ʃ sums.

I have three, they are:

1. Ʃ [ln(n)/n^2] from n=1 to ∞.

I tried the integral test but was solved to be invalid (that is, cannot divide by infinity). Therefore I believe it to be a comparison test but don't know what to compare it to.

2. Ʃ [(3n+2)/(n^3+1)] from n=0 to ∞

I solved the whole problem starting at n=1 and stating that Ʃ1/n^2 and Ʃ1/n^3 were larger and convergent, therefore the sum is convergent. I realised the n=0 part later and now cannot solve it cause you cannot divide by 0; that is, 1/0^2 or 1/0^3. I tried an index shift yet it still is invalid.

3. Ʃ [(2n+1)/n^2] from n=1 to ∞

I don't know what to compare this to. I need a value smaller than 1/n^2 which diverges (as sum diverges) but 1/n^3 converges. I don't know what to do.

Help would be much appreciated.

Thank you ]]> Differential Equations omer10000 http://www.physicsforums.com/showthread.php?t=692420 http://www.physicsforums.com/showthread.php?t=692335&goto=newpost Sat, 18 May 2013 11:37:50 GMT Does anyone know how to solve dP/dt = k P - A P2 - h for P. I understand partial fractions are needed and I have already solved dP/dt = k P - A P2.... Does anyone know how to solve dP/dt = k P - A P² - h for P. I understand partial fractions are needed and I have already solved dP/dt = k P - A P². Is anyone able to solve it, Cheers NZBRU. ]]> Differential Equations NZBRU http://www.physicsforums.com/showthread.php?t=692335 http://www.physicsforums.com/showthread.php?t=692263&goto=newpost Fri, 17 May 2013 20:52:08 GMT Hi, I am trying to find the following integral of bessel functions, any help would be great:
∫H₀(z)²/z dz

Thanks ]]> Differential Equations besselevil http://www.physicsforums.com/showthread.php?t=692263 http://www.physicsforums.com/showthread.php?t=692259&goto=newpost Fri, 17 May 2013 20:25:51 GMT [tex]\hbox { Bessel's equation of order p: }\;x^2y''+xy'+(x^2-p^2)y=0\;\hbox { and let}\;y=\sum_0^{\infty}C_nx^{n+r}[/tex]
[tex]\Rightarrow\; \left[\sum_0^{\infty}C_k(k+r)(k+r-1)x^k+\sum_0^{\infty}C_k(k+r)x^k+\sum_2^{\infty} C_{k-2}x^k-p^2\sum_0^{\infty}C_kx^k\right]x^r\;=\;0[/tex]
[tex]\Rightarrow\; x^r\left[(r^2-p^2)C_0+[(r+1)^2-p^2]C_1 x+\sum_2^{\infty}\left([(k+r)(k+r)-p^2]C_k+C_{k-2}\right)x^k\right]\;=\;0[/tex]

According to the book, all the terms have to be zero for the equation to be zero.
[tex](r^2-p^2)C_0=0\;\Rightarrow r=^+_-p[/tex]
The book also say
[tex][(r+1)^2-p^2]C_1=0\;\Rightarrow C_1=0\;\hbox { because from above, }\;r=^+_-p[/tex]


My problem with this assumption ##C_1=0## is I can just as easy say ##(r+1)^2-p^2=0## and claim ##C_0=0##!!! This will change the whole equation of the Bessel's equation!!!

Why do I have to follow the book to let ##C_1=0## which result in all ##C_{2n+1}=0##? Is it just because it is simpler and more convenient this way?

Also, the book just said let ##C_0=\frac {1}{2^p\Gamma(1+p)}## without explaining why. Why?

I know it is for fitting the formula of:

[tex]J_p=\sum_0^{\infty} \frac{(-1)^k}{k! \Gamma(k+p+1)}\left(\frac {x}{2}\right)^{2k+p} [/tex]

But can you just let ##C_0## to be anything?

Thanks ]]> Differential Equations yungman http://www.physicsforums.com/showthread.php?t=692259 http://www.physicsforums.com/showthread.php?t=692101&goto=newpost Thu, 16 May 2013 20:59:08 GMT I have a problem which in involves a second order differential equations with imaginary roots and I can seem to know how to finish the problem. ... I have a problem which in involves a second order differential equations with imaginary roots and I can seem to know how to finish the problem.

d^2y/dt^2 +15y =cos 4t+2 sin t

this is what I got so far


r^2+15=0 for the homogeneous part

r=+-(√15)

Yh=C1cos√15+C2sin√15

now is where I get stuck ]]> Differential Equations pedro123 http://www.physicsforums.com/showthread.php?t=692101 http://www.physicsforums.com/showthread.php?t=692048&goto=newpost Thu, 16 May 2013 15:40:27 GMT Greetings- In trying to solve a thermal stress problem, I have encountered an inhomogeneous differential equation of the following general form:... Greetings-
In trying to solve a thermal stress problem, I have encountered an inhomogeneous differential equation of the following general form:
[tex] \nabla^2 \Phi(r,z) = F_r(r)F_z(z)[/tex]
Solving the homogeneous case is no problem, as it is kind of a classic. Is there a route to finding a particular solution for the inhomogeneous case? Since my "charge density" (it's really temperature) is separable, I expected this to be straightforward.

It may, in fact, be straightforward, but it is still beyond my ken.

Thanks,
-BK ]]> Differential Equations badkitty http://www.physicsforums.com/showthread.php?t=692048 http://www.physicsforums.com/showthread.php?t=691930&goto=newpost Wed, 15 May 2013 22:00:22 GMT Hi, In the study of dynamical systems, phase portraits play an important role. However, in almost all related text, I only see some standard... Hi,

In the study of dynamical systems, phase portraits play an important role. However, in almost all related text, I only see some standard examples like prey-predator, pendulum etc. I have a rather unclear thought in my head regarding the role of real/imaginary eigenvalues in the system. What role do they play with respect to the system dynamics ? Are there any physical systems that you know of, which only deal with either real or imaginary eigenvalues ? Or any practical cases where one would "want" to deal with any one of those ? ]]> Differential Equations bhatiaharsh http://www.physicsforums.com/showthread.php?t=691930 http://www.physicsforums.com/showthread.php?t=691926&goto=newpost Wed, 15 May 2013 21:32:20 GMT I need help finding a linear homogenous constant-coefficient differential equation with the given general solution. y(x)=C1e^x+(C2+C3x+C4x^2)e-x ... I need help finding a linear homogenous constant-coefficient differential equation with the given general solution.

y(x)=C1e^x+(C2+C3x+C4x^2)e-x


2. I tried to come with differential equation but this is it
I can 't seem how to begin ]]> Differential Equations pedro123 http://www.physicsforums.com/showthread.php?t=691926 http://www.physicsforums.com/showthread.php?t=691610&goto=newpost Tue, 14 May 2013 03:46:47 GMT Hi Everyone! My first time to ask for help here. Waiting for any advise. I want to apply a DFT to an array of real data x_n, n=0,....,N-1 with... Hi Everyone!

My first time to ask for help here. Waiting for any advise.

I want to apply a DFT to an array of real data x_n, n=0,....,N-1 with the so-called staggered Dirichlet boundary condition that x_-1 + x_0 = 0 and x_N-1 + x_N = 0. In fact, I want to use this DFT to solve a PDE on stagger grids and with Dirichlet B.C.

Any help on this subject will be appreciated.

Thanks.

Zergon ]]> Differential Equations Zergon http://www.physicsforums.com/showthread.php?t=691610 http://www.physicsforums.com/showthread.php?t=691583&goto=newpost Mon, 13 May 2013 22:55:40 GMT My DE skills are a bit rusty, and I need some help remembering how to handle a system such as:

[itex]\dot{x_1}=x_2[/itex]
[itex]\dot{x_2}=-2x_1-3x_2+sint+e^t[/itex]

I have found the homogeneous solution to be (sorry I don't know how to do matrices here):

[itex]c_1\left\{e^{-t}\right\}+c_2\left\{e^{-2t}\right\}[/itex]
[itex]c_1\left\{-e^{-t}\right\}+c_2\left\{-2e^{-2t}\right\}[/itex]

From what I've found online I should guess a particular solution form:

[itex]x_{p}=Asin(t)+Bcos(t)+Ce^{t}[/itex]

Where A, B, and C are 2x1 matrices of constants [itex]a_1, a_2, b_1, b_2, c_1, c_2[/itex]

Is this correct?
Then rewrite the original in the form:

[itex]\dot{x_{p}}=Ax_{p}+g[/itex]

Then differentiate the guess and substitute back into the above.

Assuming this is all correct, what are the next steps in finding the general solution? ]]> Differential Equations gkirkland http://www.physicsforums.com/showthread.php?t=691583 http://www.physicsforums.com/showthread.php?t=691438&goto=newpost Mon, 13 May 2013 03:36:17 GMT dy/dt = 2 - 2ty
y(0) = 1

I am not asked to solve this (I know it's not easy to solve), but what I am asked is,

"for large values of t is the solution y(t) greater than, less than, or equal to 1/t"?

I would think less than because 1/e^(t^2) converges faster than 1/t, but at the same time I solved dy/dt = 0 and got y = 1/t, so I'm not sure what that signifies. ]]> Differential Equations Gridvvk http://www.physicsforums.com/showthread.php?t=691438 http://www.physicsforums.com/showthread.php?t=691086&goto=newpost Sat, 11 May 2013 08:02:53 GMT Solve the equation [itex]m\frac{d^{2}x}{dt^{2}} + c\frac{dx}{dt} + kx = (ax + b)^{2} + c^{2}[/itex] for the constants [itex]m, c, k[/itex]

The right hand side a, b, and c are arbitrary digits. For me they are a = 2, b = 3, and c = 8.
The problem recommends creating a linear system of equations for me to solve. This is to be done using MapleSoft.

I tried making a set of equations using (m*r^2 + c*r + k) = (2+3)^2 + 64 and then using various values of r to get a system of equations. But the result got me [itex]m = 0, c = 0[/itex] and k equaling some number. However this is incorrect because the left part of the main equations with the constants will be used to make a vibration model that I would solve. With m and c equaling 0, I would not have any derivatives and therefore not being able to solve the equation.

So now I'm at a loss. I also tried converting the equation into 2 systems of first order differentials but that lead to me really nowhere. ]]> Differential Equations Abyssnight http://www.physicsforums.com/showthread.php?t=691086 http://www.physicsforums.com/showthread.php?t=690854&goto=newpost Fri, 10 May 2013 00:49:05 GMT Hi everyone,

I am trying to solve and plot the function, x'(t) = 1 + t*sin(t*x) where x(0) = 0 and t_final = 1, in order to compare this exact solution to the approximations of Euler's and Improved Euler's Method. Can anyone help me with the code in order to solve this problem, and then plot it, using Matlab?

Thanks in advance!! ]]> Differential Equations bdoherty1994 http://www.physicsforums.com/showthread.php?t=690854