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Debnath & Mikusinski Introduction To Hilbert Spaces With by Debnath & Mikusinski

By Debnath & Mikusinski

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Let the origin w the angular Find the solution for arbitrary z'. x', y', and A motor produces a torque proportional to a control It is coupled to a fly- voltage fed into a regulator. wheel of moment of intertia which drives a load I exerting a constant retarding torque tor control voltage is proportional to is the angular speed and pression for w(t). w0 N. If the regulaw-w 0 , where a constant, find an ex- Assume that w(O) = 0. w PART I: 46 19. THE LAPLACE TRANSFORM Suppose that in Problem 16 the torque supplied by the motor is proportional to a control voltage fed into a regulator.

Express the solution to Problem 13 as a convolution if the applied voltage is an arbitrary function E(t). §3. Ordinary differential equations 16. Two flywheels of moment of inertia 45 and I1 coupled by an elastic shaft of stiffness k. I2 are The first flywheel is coupled to a motor which applies a constant t = 0, the second drives a load which ap- torque from plies a torque proportional to the angular velocity. (Neglect the moment Describe the motion of the system. ) 17. We wish to find the motion of a particle projected from a fixed point on the earth's surface.

EPX F (p} dp' ( 2 0} Consider the inte- y > c. R J: f(y} e y(x-y} sin R(x-y) dy x-y 1 e-yu sin Ru du. 11 rx f(x+u} u (22) THE LAPLACE TRANSFORM PART I: 20 If we break the integral into two, from to zero, and to become infinite, there are R zero to infinity, and allow -x three possibilities; namely, X < 0 X 0 + f(x+O)], > 0. X (23) This result is generally known as the inversion theorem for Laplace transforms, and is expressed by the reciprocal pair of equations F(p) f(x) where f(x) r 0 1 2'Tii f(x) e-px dx, r+~oo y-lOO is taken as F (p) e (24) Re(p) > c, px dp' y > c 1 z[f(x-0) + f(x+O)] ' (25) at a point of discontinuity.

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