\sum_{n=0}^{\infty} a_n t^{n+r} = \sum_{n=2}^{\infty} a_n t^{n+r}+a_0t^r+a_1t^{r+1}
Putting this, and your expansion into the D.E we can group all the sums, since they start at n=2,
\sum_{n=2}^{\infty} [4(n+r)(n+r-1)+8(n+r)(n+r+1)+1]a_nt^{n+r}+8(r+2)(r+1)a_1t^{r+1}+a_0t^r+a_1t^{r+1}=0...
Alright, well, messing with that last sum:
\sum_{n=0}^{\infty}(4a_n(n+r)(n+r-1)+a_n)t^{n+r}+8\sum_{n=1}^{\infty}a_n(n+r)(n+r-1)t^{n+r-1}+8a_0(r)(r-1)t^{r-1}=0
So the lowest power is this t^{r-1} fella, in which case r(r-1)=0 . But from the wording of later parts in this question it seems...
Hey Zondrina! Cool, so making the substituion t=x-1 the D.E then becomes
-4(t^2+2t)y''(t)-y(t)=0
and the assumed solution is
y=\sum_{n=0}^{\infty} a_n t^{n+r}
so we end up with
4\sum_{n=0}^\infty a_n (n+r)(n+r-1)t^{n+r}+8\sum_{n=0}^\infty a_n...
Homework Statement
The task is to find an analytic solution to the O.D.E
4(1-x^2)y''-y=0 \hspace{20mm} y'(1)=1
by using an appropriate series solution about x=1.
The Attempt at a Solution
The singularity at x=1 is regular, which makes me think the Frobenius method is what's meant by...
An alternative method to the one above, that gives you two equations with the same u (initial velocity) would be to use the fact that you know how far the bike has traveled after 30 seconds :)
I used the Laplacian as expressed in spherical coordinates, which i certainly agree, is not taking grad twice :smile:
That said, I suppose I could have used:
E=-2kr^{-3} \hat{r}
so:
\nabla \cdot E = \frac{\rho}{\epsilon_0}
\rho = \epsilon_0 \nabla \cdot (-2kr^{-3} \hat{r})
Now in my...
Hello Tiny Tim!
I was hoping by working in spherical coordinates I could make my life easier. My book assures me here grad is given by
\nabla t = \frac{\partial t}{\partial r}\hat{r} + \frac{1}{r}\frac{\partial t}{\partial \theta}\hat{\theta}+\frac{1}{rsin(\theta)}\frac{\partial...
Homework Statement Find the electric charge centred in a sphere of radius a, centered at the origin where the electric potential is found to be (in spherical coordinates) V(r)=kr^-2 where k is some constant.
The Attempt at a Solution
We have E=-\nabla V = -2kr^{-3} \hat{r}
So...
Homework Statement
Using the Leibniz rule and:
\nabla_{c}X^{a}=\partial_{c}X^a+\Gamma_{bc}^{a}X^b
\nabla_{a}\Phi=\partial\Phi
Show that \nabla_c X_a = \partial_c X_a - \Gamma^{b}_{ac}X_{b} .
The question is from Ray's Introducing Einsteins relativity,
My attempt...
Hello all,
Homework Statement
Given \frac{dE}{dt}=\frac{d(m(u))}{dt}\cdot u
show that \frac{dE}{dt}=\frac{m_0}{(1-u^2)^{\frac{3}{2}}}u\frac{du}{dt}
where u is velocity, m(u) is relativistic mass, and m_0 is rest mass.
Homework Equations
m(u)=\frac{m_0}{\sqrt{1-u^2}}
The...