Recent content by John 123

  1. J

    Bessel functions of the first kind

    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...
  2. J

    Bessel functions of the first kind

    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...
  3. J

    Bessel functions of the first kind

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

    Bessel functions of the first kind

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

    Bessel functions of the first kind

    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
  6. J

    Series solution for ode by undetermined coefficients

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

    Series solution for ode by undetermined coefficients

    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!}...)...
  8. J

    Taylor series for differential equation solution

    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
  9. J

    Taylor series for differential equation solution

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

    Taylor series for differential equation solution

    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
  11. J

    Taylor series for differential equation solution

    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...
  12. J

    Taylor series for differential equation solution

    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)...
  13. J

    Taylor series for differential equation solution

    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...
  14. J

    Projectile Problem: Finding Max Range & Height, Time to Reach Ground

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