Recent content by einstein314

Homework Statement Prove that: \cos^6{(x)} + \sin^6{(x)} = \frac{5}{8} + \frac{3}{8} \cos{(4x)} Homework Equations I am not sure. I used factoring a sum of cubes. The Attempt at a Solution I tried \cos^6{(x)} + \sin^6{(x)} = \cos^4{(x)} - \cos^2{(x)} \sin^2{(x)} + \sin^4{(x)} . But I...

Define y(t) = p(t) - 1 . Then \frac{d^2y}{dt^2} = \frac{d^2p}{dt^2} and the differential equation becomes: \frac{d^2y}{dt^2} = -ky Auxiliary equation is r^2 + k = 0 so r = \pm i \sqrt{k} . Then y = e^a(c_1 cos(bt) + c_2 sin(bt)) = e^{(0)}(c_1 cos(t\sqrt{k}) + c_2 sin(t\sqrt{k})) = c_1...

Homework Statement A particle moving from a point A to a point B , 1 meter away, travels in a straight line in such a way so that its acceleration is proportional to the distance left to point B . If the particle arrives at point B in 1 second, how long did it take for the particle to...

This one was a 1968 Putnam competition problem, I believe: {\int_{0}^{1}{\frac{x^4 (1-x)^4}{1+x^2} dx}} The answer is really interesting... If you're really up for a challenge, try this continuation: {\int_{0}^{1}{\frac{x^8 (1-x)^8 (25+816x^2)}{3164(1+x^2)} dx}} These are just tedious and...