Introduction : The step by step solution of the question how to Solve homogeneous linear difference equation x(n+2)-2x(n+1)+10x(n)=0,x(0)=1/2,x(1)=3/2 is explained below.
Question : Solve
\[x_{n+2}-2x_{n+1}+10x_{n}=0\]
with initial conditions \(x_0=\frac{1}{2}, x_1 =\frac{3}{2}\)
Solution:
We have given is homogeneous Linear difference equation
\[x_{n+2}-2x_{n+1}+10x_{n}=0\]
To solve a homogeneous linear difference equation we first find it’s characteristic equation
\[r^2-2r+10=0\]
\[r^2-2r+1+9=0\]
\[(r-1)^2-(3i)^2=0\]
\[(r-1-3i)(r-1+3i)=0\]
\[r_1 =1+3i, r_2 =1-3i\]
roots are complex
In complex case General solution of homogeneous Linear difference equation is of the form
\[x_{n}=\left(A\cos\left(n\tan^{-1}\left(\frac{3}{1}\right)\right)+B\sin\left(n\tan^{-1}\left(\frac{3}{1}\right)\right)\right)\]
\[x_{n}=10^{\frac{n}{2}}\left(A\cos\left(n\tan^{-1}\left(\frac{3}{1}\right)\right)+B\sin\left(n\tan^{-1}\left(\frac{3}{1}\right)\right)\right)\]
Now To find constants A and B
Put n=0 and using first initial condition \(x_0 =\frac{1}{2}\)
\[\frac{1}{2}=A\]
and Put n=1 and using second initial condition (x_1 = \frac{3}{2}\)
\[\frac{3}{2}=10^{\frac{1}{2}}\left(A\cos\left(\tan^{-1}\left(\frac{3}{1}\right)\right)+B\sin\left(\tan^{-1}\left(\frac{3}{1}\right)\right)\right)\]
Solving these equations for A, B
\[A = 1, B = \frac{\left(\frac{3}{2\cdot\sqrt{10}}-\cos\left(\tan^{-1}\left(\frac{3}{1}\right)\right)\right)}{\sin\left(\tan^{-1}\left(\frac{3}{1}\right)\right)} = \frac{1}{6}\]
Therefore specific solution of the given difference equation becomes now
\[x_{n}=10^{\frac{n}{2}}\left(\cos\left(n\tan^{-1}\left(\frac{3}{1}\right)\right)+\frac{1}{6}\sin\left(n\tan^{-1}\left(\frac{3}{1}\right)\right)\right)\]