Recent content by T-chef

\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...

Of course! The left hand side is exactly the scaler field I need. All comes out nicely after that! Thank you sir,

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...

That makes much more sense, thanks for the help guys

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...