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A mass m=1kg, initially at rest and with x=10mm, is connected to a damper with stiffness k=24N/mm and damping constant c=0.2Ns/mm. Given that the differential equation of the system is given by d^2x/dt^2+(dx/dt *c/m)+kx/m=0, find the particular solution.

The system is described by a homogeneous, second order differential equation d 2 x/dt 2 +(dx/dt * c/m) + kx/m =0. First, substitute the known constants (m,k,c) to get d 2 x/dt 2 +0.2dx/dt + 24x =0. The auxil...
CK
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Find the eigenvalues and eigenvectors of the following 3x3 matrix (reading left to right, top to bottom): (1 0 2 3 1 1 2 0 1)

The eigenvalues are given by the characteristic equation (1-x)((1-x)^2-4)=0, which gives the values x=1, x=-1 and x=3 . These eigenvalues correspond to the eigenvectors (0, 1, 0), (1, -1, -1) and (1, -5, 1) ...
JP
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Find the general solution to: d^(2)x/dt^(2) + 7 dx/dt + 12x = 2e^(-t)

"General Solution = Complimentary Function + Particular Integral"AE: m 2 + 7m + 12 = 0 solve for m (m+3)=0m = -4 or -3Hence the Complimentary Function = Ae -4t + Be -3t PI: [substitute u=ke -t ]u'=...
EO
4007 Views

Find the stationary points of the function z = 3x(x+y)3 - x3 + 24x

z = 3x(x+y) 3 - x 3 + 24xDifferentiating partially with respect to x and with respect to y:∂z/∂x = 3(x+y) 3 + 9x(x+y) 2 - 3x 2 + 24∂z/∂y = 9x(x+y) 2 At stationary points: ∂z/∂x = 0 and ∂z/∂y = 0.From ∂z/∂y =...
HT
2574 Views

The rectangular hyperbola H has parametric equations: x = 4t, y = 4/t where t is not = 0. The points P and Q on this hyperbola have parameters t = 1/4 and t = 2 respectively. The line l passes through the origin O and is perpendicular to the line PQ.

This question asks us to find the cartesian equation of l. First we must find the points P and Q. To do this we substitute t with 1/4 to find P and substitute t with 2 to find Q.Doing this we get the coordin...
LB
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