Using the third order Taylor’s series method, find the solution of the initial value problem:
at x = 0.1 taking h = 0.1
To solve the initial value problem using the third-order Taylor series method, we consider the differential equation in the form $y' = f(x, y)$ with an initial condition $y(x_0) = y_0$. The third-order Taylor series formula is given by $$ y_{n+1} = y_n + h y'_n + \frac{h^2}{2} y''_n + \frac{h^3}{6} y'''_n, $$ where $y'_n = f(x_n, y_n)$, ________ _________ ____ __________ ________ ________ _____ __________ ____ _____ _______ _____.
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Check whether or not (3) is a maximal ideal in Z9.
Find the order of each of the elements in U(15). Is U (15) cyclic? Justify your answer.
Which of the following statements are true and which are false? Give a short proof or a counterexample in support of your answer:
i) The equation x3 4x 16 = 0 has a root in the interval [3, 4].
ii) The order of convergence of the secant method is 0.62.
iii) For the system of linear equations:
5x + y + 2z = 34
4y − 3z = 12
10x − 2y + z = −4
the matrix is diagonally dominant.
iv) The numerical method:
is relatively stable.
v) The method:
converges to 1.5 for any choice of initial approximation.
Use the Euler’s method to solve numerically the initial value problem:
with h = 2.0 on the interval [0, 1].
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Show that, if G is a finite group and a ∈ G, o(a) | o(G). Further, show that a°(G) = e for all a ∈ G. Deduce the Euler-Fermat theorem a(n) = 1 (mod n) for all a, n ∈ N, n ≥ 2, (a, n) = 1.
If F is a field, show that U(F[x]) = F.