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Find the general solution of the second order differential equation y''(t)+y(t) = 5exp(2t)

The general solution y GS (t) to a differential equation in the form ay''(t)+by'(t)+cy(t)=f(t) is the sum of the complementary function y CF (t) and a particular solution y PS (t).First we find the complemen...
AC
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Prove by induction the sum of the natural numbers from 1 to n is n(n+1)/2

Need whiteboard throughout to properly answer, so will go through the ideas of what to do: Take the base case of when n=1, and show that (sum from 1 to 1) j = n(n+1)/2 is equal to 1. Take the assumption that...
OO
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Simplify i^{4}?

Simplify i 4 ? We know that i = -1 1/2 Therefore i 2 =-1 As i 4 =i 2 xi 2 Then i 4 =-1x-1=1
HS
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The infinite series C and S are defined C = a*cos(x) + a^2*cos(2x) + a^3*cos(3x) + ..., and S = a*sin(x) + a^2*sin(2x) + a^3*sin(3x) + ... where a is a real number and |a| < 1. By considering C+iS, show that S = a*sin(x)/(1 - 2a*cos(x) + a^2), and find C.

C + iS = (a cos(x) + a^2 cos(2x) + a^3 cos(3x) + ...) + i( a sin(x) + a^2 sin(2x) + a^3 sin(3x) + ...) = a(cos(x) + i sin(x)) + a^2(cos(2x) + i sin(2x)) + a^3(cos(3x) + i*sin(3x)) + ... = a e^ix + a^2 e^i2x ...
SF
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Prove by induction that the sum of the first n integers can be written as (1/2)(n)(n+1).

For n = 1, the sum is given by (1/2)(1)(1+1), which gives 1, the expected result. We now assume that the statement is true for some k. If we look at k+1, the sum is given by 1 + 2 + ... + k + (k+1). Since we...
JS
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