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What is the polar form of the equation: x^2+y^2 =xy+1

Using Pythagoras, x 2 + y 2 = r 2 .Using basic trigonometry, x = rsinθ and y = rcosθ. xy + 1 = r 2 sinθcosθ + 1 = (1/2)r 2 sin2θ + 1 Subbing in both halves and doubling gives:2r 2 = r 2 sin2θ + 2 -> r 2 (...
HW
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Give the general solution to y'' - 3y' + 2y = 4x

note: y' is often written as dy/dx, this doesn't change the question at all so no need to worry. Same for y'' and d 2 y/dx 2 .All second-order differential equations that don't have 0 on the right hand side ...
DS
7458 Views

Let A, B and C be nxn matrices such that A=BC-CB. Show that the trace of A (denoted Tr(A)) is 0, where the trace of an nxn matrix is defined as the sum of the entries along the leading diagonal.

This is the type of question which requires some out of the box thinking. A sketch is presented below: We start by defining A = (a ij ), B = (b ij ) and C = (c ij ). We then use the definition of matrix mult...
DG
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Express sin(5theta) in terms of sin(theta) and powers of sin(theta) only.

Consider the expression (cos(theta) + i*sin(theta)) 5 . (Where theta a real parameter).By De Moirve's theorem, we know this expression is equivalent to cos(5theta) + i sin(5theta).We can also apply the binom...
PF
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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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