sinx+sin3x+sin(√x)=0, then find the general solution?


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Question Number 27843 by bmind4860 last updated on 15/Jan/18

$s i n x + s i n 3 x + s i n \sqrt{x} = 0, t h e n f i n d t h e$ $g e n e r a l s o l u t i o n ?$
Commented by abdo imad last updated on 17/Jan/18

$l e t p u t \sqrt{x} = t e \Leftrightarrow s i n t + s i n (t^{2}) + s i n (3 t^{2}) g e n e r a l c a s e$ $s i n t + \sum_{k = 0}^{n} s i n (2 k + 1) t^{2} = 0 a l s o w e c a n g e t g e r a l c a s e$ $\sum_{k =}^{n} s i n (2 k + 1) x + s i n (\sqrt{x}) = 0 f o r e x a m p l e l e t f i n f$ $\sum_{k = 0}^{n} s i n (2 k + 1) t^{2} = \sum_{k = 0}^{n} s i n (2 k + 1) u w i t h u = t^{2}$ $a n d \sum_{k = 0}^{n} s i n (2 k + 1) u = I m (\sum_{k = 0}^{n} e^{i (2 k + 1) u})$ $= I m (e^{i u} \sum_{k = 0}^{n} {(e^{i (2 u)})}^{k}) = I m (e^{i u} \frac{1 - {(e^{i (2 u)})}^{n + 1}}{1 - e^{i 2 u}}) i f$ $e^{i (2 u)} \neq 1 . . . . . .$
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