DE#6

Dec 2015
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Earth
(1) \(\displaystyle \: y''(1+yy')=y'(1+y'^2 )\).

(2) \(\displaystyle \: y''-y'+e^{4x}y=0\).
 

skipjack

Forum Staff
Dec 2006
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(2) $y'' - y' + e^{4x}y = 0$
Let $y = e^{x/2}u$, then the equation becomes $u'' + (e^{4x} - 1/4)u = 0$.
Let $(1/2)e^{2x} = t$, then the equation becomes $t^2d^2u/dt^2 + tdu/dt + (t^2 - 1/4^2)u = 0$,
whch is a standard equation whose general solution is a linear combination of Bessel functions.
 
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Dec 2015
1,085
169
Earth
(1) \(\displaystyle y''+yy'y''=y'+y'^{3} \: \) or \(\displaystyle \: dy'+yy'dy'=dy+y'^{2} dy\).

\(\displaystyle y'-y=c+ 2\int y'^{2} dy -\int y'd(yy')\).
how to continue ?
 
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