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}^{T}a_{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\}\,

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