) Using fourth order Taylor series method with , solve the initial value problem
upto .
Given (y' = f(x,y) = x + \cos y,\quad y(0)=0,\quad h=0.2). For fourth order Taylor method, [ y_{n+1}=y_n+h y'_n+\frac{h^2}{2}y''_n+\frac{h^3}{6}y'''_n+\frac{h^4}{24}y''''_n. ] First compute derivatives. [ y' = x+\cos y ] [ y'' = 1-\sin y; ___ _________ _________ ___ ____ _______.
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Solve the heat conduction equation in the region
with the initial and boundary conditions
using Crank-Nicolson method with
and
upto two time steps.
Based on the third image provided, here is the transcription of the remaining mathematical problems:
Using the transformation find the solution of the equation
in terms of Bessel's functions.
) If km and kn are distinct roots of Bessel function with
then show that
Find an approximate value of y(1.0) for the initial value problem
y = x - 2y, y(0) = 1
using Milne-Simpson’s method
with the step length . Calculate the starting value using Runge-Kutta fourth order method with the same h.
) For the following differential equation locate and classify its singular points on the x-axis
i)
ii)
Show that .
If the Fourier cosine transform of f(x) is , then show that