Recent content by John 123

Thanks Dick John

Hi Dick I need to show that if: u(x)=x^{\frac{1}{2}}J_k(r_ix) Then: u'=r_ix^{\frac{1}{2}}J'_k(r_ix)+\frac{1}{2}x^{\frac{-1}{2}}J_k(r_ix) This, of course, uses product rule but the derivative of: J_k(r_ix) I am unclear about? Incidentally r_i is the ith distinct positive...

Hi Dick The question I am asking derives from proving the integral property of Bessel functions of the first kind. This amounts to showing the orthogonal properties of Bessel functions. Part way through the proof I need to show that...

Then that is my error. To find the derivative of J_k(ax) presumably one has to differentiate the series expansion? John

But doesn't J_k(ax)=aJ_k(x)?

Homework Statement Can anyone tell me if: \frac{d}{dx}J_k(ax)=aJ'_k(x) where a is a real positive constant and J_k(x) is the Bessel function of the first kind. Regards John Homework Equations The Attempt at a Solution

Hi Clamtrox I believe my error is in expanding siny about y=0 instead of about y=pi/2. Regards John

Homework Statement Obtain the Taylor series solution up to and including order 3 of the following non linear ode y'=x^2+\sin y,y(0)=\frac{\pi}{2} Homework Equations After substituting the power series form of sin(y) I get: y'=x^2+(y-\frac{y^3}{3!}+\frac{y^5}{5!}-\frac{y^7}{7!}...)...

Yes my series solution agrees with the book. I have done the previous and following examples in the book using their same method and have agreed both the series solutions and their intervals of convergence. John

Hi Mark What are we concluding about this problem. Is the book answer for the interval of convergence =0.04 correct? John

Yes it is a series solution for y'=f(x,y) By the way your solution -0.42 is not consistent in terms of |x-1|, since modulus should be positive? John

Hi Mark The book is using the following theorem: If f(x,y) is analytic at (x_0,y_0), i.e if f(x,y) has a Taylor series expansion in powers of (x-x_0) and (y-y_0), valid for |x-x_0|<r,|y-y_0|<r, and if for every (x,y) in this 2rx2r rectangle which has (x_0,y_0) at its center, |f(x,y|\leq M...

Yes. In fact I have derived the nth term as: \frac{2(-1)^{n+1}(x-1)^{n+1}}{n^2+n} I then applied the RATIO TEST but was unable to agree the book answer. In fact I have been following the book method of deriving the interval of convergence by expanding: x^2-y^2 In powers of (x-1)...

Homework Statement Find the series solution for: y'=x^2-y^2,y(1)=1 Homework Equations The Attempt at a Solution I have correctly derived the series solution as: y(x)=1+(x-1)^2-\frac{(x-1)^3}{3}+\frac{(x-1)^4}{6}-... But I cannot get the book solution for the INTERVAL OF...

y_0 is the height above ground level from which the projectile is fired. John