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Find the solution the the differential equation d^2y/dx^2 + (3/2)dy/dx + y = 22e^(-4x)

We first find the complementary function by guessing y=e^(kx). Substituting this into the equation d^2y/dx^2 + (3/2)dy/dx + y = 0. we find k^2 + (3/2)k + 1 = 0 which factorises into (k+2)(k+1/2). So our comp...
NE
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Given sinhx = 0.5(e^x - e^-x), express its inverse, arcsinhx in terms of x.

The first step is to write sinhx in its exponential form and set it equal to y, this will make rearranging easier. Then multiply everything by e^x and rearrange to form a quadratic, in terms of e^x. Express ...
KA
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The finite region bounded by the x-axis, the curve with equation y = 2e^2x , the y-axis and the line x = 1 is rotated through one complete revolution about the x-axis to form a uniform solid. Show that the volume of the solid is 2π(e^2 – 1)

The volume of revolution, V, is given as 2π∫ydx Substituting in the equation and limits gives as follows: V = 2π∫2e^2x dx between 0 and 1 Integrating this gives V = 2π[e^2x] between 0 and 1 Applying the limi...
MC
7896 Views

How do I find the vector/cross product of two three-dimensional vectors?

Finding the cross product of two 3D vectors uses a very similar method to that of finding the determinant of a 3x3 matrix. So, to find the cross product, just follow these simple steps: To begin, create a 3x...
LM
7304 Views

Find the general solution for the determinant of a 3x3 martix. When does the inverse of this matrix not exist?

Let M be a 3x3 matrix s.t. M= |a b c| |g h i| |d e f| Then Det(M)= a(Det(e,f,h,i))-b(Det(d,f,g,i))+c(Det(d,e,g,h). Given that the determinant of a 2x2 matrix such as (e,f,h,i) is = ei-fh. The solution is; De...
OD
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