Question

GIVE THE COMPLETE SOLUTION,FOR THE SECOND ORDER IVP :
S(n)=2S(n-1)-S(n-2) S(0)=1,S(1)=4, (WHERE n>=2)

Answer 1

write out a few terms...
you'll see you get 3k+1

Answer 2

Do you need a formal solution?

Answer 3

yes pls

Answer 4

\[S_n=2S_{n-1}-S_{n-2}\]
The characteristic equation and roots are:
\[r^2=2r-1\\
r^2-2r+1=0\\
(r-1)^2=0\\
r=1\]
Thus, the solution is of the form:
\[S_n=\alpha_1 1^n+n\alpha_21^n=\alpha_1+n\alpha_2\]
Plugging in the initial values, we get:
\[\begin{cases}
S_0=1=\alpha_1\\
S_1=4=\alpha_1+\alpha_2
\end{cases}\\
\alpha_2=3\]
So the solution is S(n)=1+3n.