Number roots with positive real parts

Use the Routh-Hurwitz eriterion to determine the number of roots with positive real parts in the following eharaeteristie equations... 

It is often difficult to determine quickly the roots of the characteristic equation. Hurwitz(I,) and Routh( 0) developed an algebraic procedure for finding the number of roots with positive real parts and consequently whether the system is unstable or not. 

Routh-Hurwitz criterion The number of roots with positive real parts of a real polynomial equation is the number of sign changes in the following sequence ... 

Then, shift the (m-f- l)-th row to the left over k positions, so that the element rm+i,k+i becomes the first one in the line, and multiply all other entries of this row through by (-1). Since the first entry is now non-zero, one proceeds as in the regular case. Eventually, the number of roots of S(A) with positive real parts will be equal to the number of sign changes in the first column added to the sum of deficiency numbers over all irregular rows. 

If the roots are, however, complex numbers, with one or two positive real parts, the system response will diverge with time in an oscillatory manner, since the analytical solution is then one involving sine and cosine terms. If both roots, however, have negative real parts, the sine and cosine terms still cause an oscillatory response, but the oscillation will decay with time, back to the original steady-state value, which, therefore remains a stable steady state. 

The Routh criterion states that in order to have a stable system, all the coefficients in the first column of the array must be positive definite. If any of the coefficients in the first column is negative, there is at least one root with a positive real part. The number of sign changes is the number of positive poles. 

It is important to choose the branch of the square root with a positive real part in the integrations above. Summation over quantum numbers k proceeds as before with the result for the energy density that... 

To utilize this equation, we specify 81,82, and Ca, and then calculate the roots a that satisfy the equation for each value of the wave number a. If any of the roots has a positive real part, the system is unstable. If the real parts of all roots are negative, on the other hand, the system is stable. The wave number with the largest real part for a is the fastest-growing infinitesimal disturbance. 

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