An Introduction to Information Theory: Symbols, Signals and by John R. Pierce

By John R. Pierce

Covers encoding and binary digits, entropy, language and which means, effective encoding and the noisy channel, and explores ways that info conception pertains to physics, cybernetics, psychology, and paintings. "Uncommonly good...the such a lot pleasant dialogue to be found." - medical American. 1980 variation.

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111) General solution: here C1 and C2 are arbitrary constants. 112) Check: Using Maple dsolve command. 3: We are given one solution to the Cauchy-Euler homogeneous differential equation shown here. From this single basis vector, we are asked to find a complete solution to the diffferential equation. 121) yp (t) := > y[p](t):=subs(s=t,value(%)); General solution: here C1 and C2 are arbitrary constants. 122) This solution is valid everywhere except at the origin. Check: Using Maple dsolve command.

The equation, written in standard form, reads a2 d2 d y(t) + a1 y(t) + a0 y(t) = f(t) 2 dt dt Since the order of the differential equation is two, we must first find a set of two basis vectors of the corresponding homogeneous differential equation a2 d d2 y(t) + a1 y(t) + a0 y(t) = 0 dt dt 2 Since the coefficients a2, a1, and a0 are time invariant (constants), the method of undetermined coefficients can be used to find a set of system basis vectors. 32) It can be shown that if the discriminant given here is not equal to 0, then the roots r1 and r2 are distinct and the two solutions are linearly independent; thus, for constant coefficient second-order differential equations, the preceding solution vectors constitute a set of system basis vectors.

The guess, of course, depends on the character of the source function f(t) and whether or not this function is linearly dependent on the system basis vectors. With the preceding method, there is no need to guess, and we get immediate solutions. 1, we found a set of basis vectors to be y1(t) = e−t and y2(t) = te−t We note the linear dependence of the source term on one of the basis vectors. 92) Integrating first with respect to the dummy variable s and then substituting t for s yields yp (t) = t 3 e−t 6 Thus, the general solution to the problem is the sum of the homogeneous plus the particular solution.

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