Question

how to solve xy''+2y'+xy=0

Answer 1

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Answer 2

http://www.wolframalpha.com/input/?i=xy%27%27%2B2y%27%2Bxy%3D0

Answer 3

\[xy''+2y'+xy=0\]\[(xy)'=y+xy'\]\[(xy)''=(y+xy')'=y'+y'+xy''=2y'+xy''\]
\[(xy)''+xy=0\quad,\quad z=xy\quad,\quad z''+z=0\]\[z=C_1\cos{x}+C_2\sin{x}\quad\Rightarrow\quad y=\frac{z}{x}=\frac{C_1\cos{x}+C_2\sin{x}}{x}\]