According to the Gf Simmons book, the local truncation error in fourth order Runge-Kutta is $(1/180)y^{(5)}(a)h^5$, but according to me it is $(1/3840)y^{(5)}(a)h^5$
I used Taylor expansion to do that,i am attaching the picture of that part of book as well as my method.Thanks in advance for helping me with enter image description herethe doubt.
This is both wrong, the leading error term in RK4 is not a pure derivative, not a multiple of the 5th derivative.
Expressed in combinations of the ODE function, the exact 5th derivative is $$ \frac{y^{(5)}(x)}{5!} = \frac{f_{4}[f_{0}^4]}{120} + \frac{f_{3}[f_{1}[f_{0}],f_{0}^2]}{20 } + \frac{f_{2}[f_{2}[f_{0}^2],f_0]}{30} + \frac{f_{2}[(f_{1}[f_{0}])^2]}{40} + \frac{f_{2}[f_{1}[f_{1}[f_{0}]],f_{0}]}{30} + \frac{f_{1}[f_{1}[f_{1}[f_{1}[f_{0}]]]]}{120} + \frac{f_{1}[f_{1}[f_{2}[f_{0}^2]]]}{120} + \frac{f_{1}[f_{3}[f_{0}^3]]}{120} + \frac{f_{1}[f_{2}[f_{1}[f_{0}],f_{0}]]}{40} $$ where $f_0=f(x)$, $f_k=f^{(k)}(x)$, the derivative tensors, are vector-valued symmetric multilinear forms, the arguments in $[,]$ are the vector entries to these multilinear forms, powers here indicate repetition of the argument.
The first error term of RK4 in contrast is $$ \frac{y_1-y(x_1)}{h^5} = \frac{f_{4}[f_{0}^4]}{2880} + \frac{f_{3}[f_{1}[f_{0}],f_{0}^2]}{480} - \frac{f_{2}[f_{2}[f_{0}^2],f_0]}{480} + \frac{f_{2}[(f_{1}[f_{0}])^2]}{160} + \frac{f_{2}[f_{1}[f_{1}[f_{0}]],f_{0}]}{120} - \frac{f_{1}[f_{1}[f_{1}[f_{1}[f_{0}]]]]}{120} + \frac{f_{1}[f_{1}[f_{2}[f_{0}^2]]]}{480} - \frac{f_{1}[f_{3}[f_{0}^3]]}{720} - \frac{f_{1}[f_{2}[f_{1}[f_{0}],f_{0}]]}{240} $$ Alone that there are mixed signs shows that this can not be a multiple of $y^{(5)}$.
For the order conditions expressed in full partial derivatives see Prove that Runge Kutta Method (RK4) is of Order 4. The terms were computed using manual post-processing of the results of Analytical expressions for ODE integration methods using Sympy