Question

use the method of diagonalization to obtain the general solution of the given system of differential equation d^2x1/dt^2 =5x1+4x2 and d^2x2/dt^2=x1+2x2

Answer 1

Solution

Let vector X = (x₁, x₂)^T , and matrix $A=\left(\begin{matrix}5&4\\1&2\\\end{matrix}\right)$ .

Then system d²X/dt² =AX.

det(A-λI) = (5-λ)(2-λ)-4 = λ²-7λ+6 = 0 =>  λ₁=1, λ₂=6

Matrix A may be reduced to diagonal matrix $D=\left(\begin{matrix}1&0\\0&6\\\end{matrix}\right)$ by matrix S with columns, which are eigenvectors of systems (A - I)v₁=0 and (A - 6I)v₂=0.

Solving these systems we’ll get $S=\left(\begin{matrix}1&4\\-1&1\\\end{matrix}\right)$  and S^-1AS=D

The system d²X/dt² =AX is simplified by substitution Y=S^-1X to the system d²Y/dt² =DY with diagonal matrix D.

So $y_1=ae^t+be^{-t}$, $y_2=ce^{t\sqrt{6}}+de^{-t\sqrt{6}}$, where a,b,c,d – arbitrary constants.

X=SY => $x_1=ae^t+be^{-t}+4(ce^{t\sqrt{6}}+de^{-t\sqrt{6}})$ , $x_2=-ae^t-be^{-t}+ce^{t\sqrt{6}}+de^{-t\sqrt{6}}$