5.4

The complementary solution for each method is

\[u_n=c_0\left(\lambda h+\frac{1}{2}\lambda^2h^2\right)^n\quad\text{(RK2)}\]

\[u_n=c_1\left(\lambda h+\sqrt{1+\lambda^2h^2}\right)^n+c_2\left(\lambda h-\sqrt{1+\lambda^2h^2}\right)^n\quad\text{(LF)}\]

at time \( t=0 \)
\[u_0=c_0\quad\text{(RK2)}\quad\text{and}\quad \underline{u_0=c_1+c_2}\quad\text{(LF)}\]

at the first time step \( t=h\ (n=1) \)
\[u_1=\underline{u_0\left(\lambda h+\frac{1}{2}\lambda^2h^2\right)= c_1\left(\lambda h+\sqrt{1+\lambda^2h^2}\right)+c_2\left(\lambda h-\sqrt{1+\lambda^2h^2}\right)}\]

The system of the two underlined equations are solved to find \( c_1 \) and \( c_2 \)

\[c_1=\frac{1+\frac{1}{2}\lambda^2h^2+\sqrt{1+\lambda^2h^2}}{2\sqrt{1+\lambda^2h^2}}u_0\]

\[c_2=\frac{\sqrt{1+\lambda^2h^2}-\frac{1}{2}\lambda^2h^2-1}{2\sqrt{1+\lambda^2h^2}}u_0\]

Using the binomial expansion for

\[\sqrt{1+\lambda^2h^2} = 1+\frac{1}{2}\lambda^2h^2-\frac{1}{8}\lambda^4h^4+\cdots\]

\[\frac{1}{\sqrt{1+\lambda^2h^2}}= 1-\frac{1}{2}\lambda^2h^2 +\frac{3}{8}\lambda^4h^4+\cdots\]

and substituting

\[c_1=\frac{1}{2}\left(2+\lambda^2h^2-\frac{1}{8}\lambda^4h^4+\cdots\right)\left(1-\frac{1}{2}\lambda^2h^2 +\frac{3}{8}\lambda^4h^4+\cdots\right)u_0\]

\[c_2=\frac{1}{2}\left(-\frac{1}{8}\lambda^4h^4+\cdots\right)\left(1-\frac{1}{2}\lambda^2h^2 +\frac{3}{8}\lambda^4h^4+\cdots\right)u_0\]

multiplying through and combining terms leaves the following first different (underlined) terms

\[c_1=u_0+\underline{\frac{1}{16}\lambda^4h^4u_0}\]

\[c_2=\underline{-\frac{1}{16}\lambda^4h^4u_0}\]