Please provide the answers and the code.

Perform an error analysis to compare $($a$)$ vs$.($c$),$ and $($b$)$ vs$.($d$)$

procedure Taylor $4(f,a,b,x_a,n)$

integer $k$; real $h,t,x,x^{\text{'}},x^{\text{'}\text{'}},x^{\text{'}\text{'}\text{'}},x^{(4)}$

external function $f_1,f_2,f_3,f_4$

hlarr$\frac{b-a}{n}$

tlarra

$xlarr{x}_a$

output $0,t,x$

for $k=1$ to $n$ do

$x^{\text{'}}larr{f}_1(t,x)$

$x^{\text{'}\text{'}}larr{f}_2(t,x,x^{\text{'}})$

$x^{\text{'}\text{'}\text{'}}larr{f}_3(t,x,x^{\text{'}},x^{\text{'}\text{'}})$

$x^{(4)}larr{f}_4(t,x,x^{\text{'}},x^{\text{'}\text{'}},x^{\text{'}\text{'}\text{'}})$

xlarrx$+h[x^{\text{'}}+\frac{1}{2}h[x^{\text{'}\text{'}}+\frac{1}{3}h[x^{\text{'}\text{'}\text{'}}+\frac{1}{4}h[x^{(4)}]]]]$

tlarrt$+h$

output $k,t,x$

end for

procedure $RK4(f,a,b,x_a,n)$

integer $j,n$:$,$ real $K_1,K_2,K_3,K_4,a,b,x_a,h$

external function $f$

hlarr$\frac{b-a}{n}$

tlarra

$xlarr{x}_a$

output $0,t,x$

for $j=1$ to $n$ do

$K_1$larrhf$(t,x)$

$K_2$larrhf$(t+\frac{1}{2}h,x+\frac{1}{2}{K}_1)$

$K_3$larrhf$(t+\frac{1}{2}h,x+\frac{1}{2}{K}_2)$

$K_4$larrhf$(t+h,x+K_3)$

xlarrx$+\frac{1}{6}(K_1+2K_2+2K_3+K_4)$

tlarrt$+h$

output $j,t,x$

end for

end procedure $RK4$

Legend:

$f$ is the ODE function to be solved $($i$.$e$.x^{\text{'}}(t)=$dots $)$

$a$ is starting time, $x_a$ the initial condition at that time $($i$.$e$.x(a))$

$b$ is the time for which $x(b)$ must be solved

$n$ is the number of time steps

Use single $($float$)$ precision for manual calculations $($i$.$e$.6$ significant digits$)$