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Fun Function (Posted on 2010-11-23) Difficulty: 4 of 5
f''(x) + tan(x) * f'(x) + (cos(x))^2 * f(x) = 0

What is f(x)?
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note: f'(x) and f''(x) are the first and second derivatives of f(x)

No Solution Yet Submitted by Larry    
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full solution Comment 1 of 1

differential equations of the form
y'' + p(x)*y' + q(x)*y = 0
can be solved with substitutions
ln(y) = ln(z) - (1/2)*integral(p(x))

so we have
(1) ln(y) = ln(z) + (1/2)*ln(cos(x))
with some work we get
z'' + q(x)*z = 0
where q(x) = Q(x) - (1/2)*p'(x) - (1/4)*p(x)^2
which gives us

z=c1*exp(sqrt(cos(x)^2-1))/sqrt(cos(x)) - c2*exp(sqrt(cos(x)^2-1))/(2*sqrt(cos(x)))
using e^(ix) = cos(x) + i*sin(x)
and substituting into (1) above, we get
y = c1*cos(sin(x)) + c2*sin(sin(x))


  Posted by Daniel on 2010-11-23 17:46:20
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