ODELS8

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For the differential equation $u''+a_1(t)u'+a_2(t)u=0\,$ , where $a_i(t+T)=a_i(t)\,$ for all $t (i=1,2)\,$ , show that the characteristic multipliers $\rho_1\,$ and $\rho_2\,$ satisfy the relation $\rho_1\rho_2 = \exp\left\{-\int_0^T a_1(t) dt \right\}\,$ .

Let $x_1 = u, x_2=u'\,$ . Now the equation can be written as a first order system:

$x_1'=x_2\,$

$x_2'=-a_1(t)x_2-a_2(t)x_1\,$

$A = \begin{bmatrix} 0 & 1 \\ -a_2(t) & -a_1(t) \\ \end{bmatrix}$

Assume we have two linearly independent solutions $u^1\,$ and $u^2\,$ . Now a fundamental matrix is

$X = \begin{bmatrix} u^1 & u^2 \\ u'^1 & u'^2 \\ \end{bmatrix}$

Also assume that $u^1(0)=1, u^2(0)=0, u'^1(0)=0, u'^2(0)=1\,$ .

Now the monodromy matrix is

$B = \begin{bmatrix} u^1(T) & u^2(T) \\ u'^1(T) & u'^2(T) \\ \end{bmatrix}$

The determinant of $B\,$ is $\exp\left\{\int_0^T \mathrm{tr}A\right\} = \exp\left\{-\int_0^Ta_1(t)dt\right\}\,$ .

The characteristic multipliers $\rho\,$ are the eigenvalues of $B\,$ .

$\rho^2-2\phi\rho+exp\left\{-\int_0^T a_1(t)dt \right\}\,$

Therefore $\rho_1\rho_2 = \exp\left\{-\int_0^T a_1(t) dt \right\}\,$

Main Page : Ordinary Differential Equations : Linear Systems

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