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Find the complementary function to the second order differential equation d^2y/dx^2 - 5dy/dx + 6x = x^2

Use the auxiliary equation k 2 -5k+6=0. Solving this gives roots k=2 and k=3, which are real and distinct roots. This means that the complementary function is of the form y=Ae^(k 1 x)+Be^(k 2 x), where k 1 a...
SM
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Prove by induction that the sum from r=1 to n of (2r-1) is equal to n^2.

Firstly, we note that there are 4 key steps for any proof by induction. (1) Basis : check the statement works for n=1. (2) Assumption : assume the statement is true for n=k. (3) Induction : prove the stateme...
JL
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f(x) = 2x^3+6x^2-18x+1. For which values of x is f(x) an increasing function?

For f(x) to be an increasing function at x, the derivative of f(x) must be greater than 0. So, the answer to this question will be revealed by solving the equation f'(x) > 0 for x.By differentiating the t...
NW
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A particle is moving in a straight line with simple harmonic motion. The period of the motion is (3pi/5)seconds and the amplitude is 0.4metres. Calculate the maximum speed of the particle.

Question: A particle is moving in a straight line with simple harmonic motion. The period of the motion is (3 pi/5)seconds and the amplitude is 0.4 metres. Calculate the maximum speed of the particle. Anwer:...
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Show that the square of any odd integer is of the form (8k+1)

Any odd integer can be represented by 2m+1 for some integer m (can prove that 2m+1 is always an odd number if needed). (2m+1) 2 =4m 2 +2m+2m+1=4m 2 +4m+1=4m(m+1)+1 simple factorisation. m and (m+1) are conse...
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