Question
b) The differential equation of a damped vibrating system under the action of an external periodic force is:
Show that, if the complementary function of the differential equation represents vibrations which are soon damped out. Find the particular integral in terms of periodic functions.
Answer :
Word Count : 601
The given differential equation is: \[ \frac{d^2x}{dt^2} + 2m_0 \frac{dx}{dt} + n^2 x = a \cos(pt) \] ### Step 1: Solving the Homogeneous Equation (Complementary Function) First, we solve the homogeneous equation (without the external force term): \[ \frac{d^2x}{dt^2} + 2m_0 \frac{dx}{dt} + n^2 x = 0 \] This is a second-order linear differential equation with constant coefficients. The characteristic equation associated with this is: \[ r^2 + 2m_0 r + n^2 = 0 \] Using the quadratic formula to solve for \( r \): \[ r = \frac{-2m_0 \pm \sqrt{(2m_0)^2 - 4 \cdot 1 \cdot n^2}}{2 \cdot 1} \] ________ _____ ________ ________ ____ ______ _________ _________ __________ __________.
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The given differential equation is: \[ \frac{d^2x}{dt^2} + 2m_0 \frac{dx}{dt} + n^2 x = a \cos(pt) \] ### Step 1: Solving the Homogeneous Equation (Complementary Function) First, we solve the homogeneous equation (without the external force term): \[ \frac{d^2x}{dt^2} + 2m_0 \frac{dx}{dt} + n^2 x = 0 \] This is a second-order linear differential equation with constant coefficients. The characteristic equation associated with this is: \[ r^2 + 2m_0 r + n^2 = 0 \] Using the quadratic formula to solve for \( r \): \[ r = \frac{-2m_0 \pm \sqrt{(2m_0)^2 - 4 \cdot 1 \cdot n^2}}{2 \cdot 1} \] ________ _____ ________ ________ ____ ______ _________ _________ __________ __________.
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