math245lab07 - Math245 Computer Lab set#7 Fall 2008 Roots...

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Math245 Computer Lab set #7, Fall 2008 Roots of characteristic equation In Matlab, there is a very useful command roots to find all the roots of the polynomial. By using roots , you can obtain the roots of characteristic equation of linear higher order ODE, and also you can determine whether the solution of ODE is stable or unstable. Example 1 Consider the 4rd-order linear, homogeneous ODE y (4) + 3 5 y 000 + 138 100 y 00 + 231 500 y 0 + 2929 10000 y = 0 . (1) Noting y e rt , the characteristic equation of this ODE is r 4 + 3 5 r 3 + 138 100 r 2 + 231 500 r + 2929 10000 = 0 . We can find the roots of this characteristic (polynomial) equation very easily as shown below. EDU>> roots([1 3/5 138/100 231/500 2929/10000]) ans = -0.1000 + 1.0000i -0.1000 - 1.0000i -0.2000 + 0.5000i -0.2000 - 0.5000i Roots of the characteristic equation are r 1 , 2 = - 0 . 1 ± i , and r 3 , 4 = - 0 . 2 ± 0 . 5 i . The general solution is written y = e - 0 . 1 t { A cos t + B sin t } + e - 0 . 2 t { C cos 0 . 5 t + D sin 0 . 5 t } . Since the real part of characteristic roots are all negative, the solution will die out for large t hence this system is stable.
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