
#1
Aug2005, 06:33 AM

P: 60

Q1. Let u(x) be a function with the property that the area under the curve between any two points, a, b with a<b, is directly proportional to the different of the functional values at a and b. Obtain a differential equation for u(x).
Q2. A parabolic reflector has the property that a light source placed at its focus produces a parallel beam, or, conversely, parallel rays converge at the focus. Assuming that reflection of light from a curve is determined by the usual laws of reflection for the tangent to the curve at the point of incidence (angle of incidence equals angle of reflection), use the above property to determine a differential equation for the parabola. 



#2
Aug2005, 10:40 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,881

In order to get a differential equation, fix a and take b= x as variable. The "area under the curve" is now [itex]\int_a^x u(t)dt[/itex] and that must be equal to u(x) u(a): [itex]\int_a^x u(t)dt= u(x) u(a)[/itex]. What do you get if you differentiate both sides of that equation? Of course, m is [itex]\frac{dy}{dx}[/itex]. 



#3
Aug2105, 01:52 AM

P: 60

thx for ur help!



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