solve-x-3-d-3-y-dx-3-2x-2-d-2-y-dx-2-2y-10-x-1-x-

Question Number 169287 by cortano1 last updated on 28/Apr/22

s o l v e : x^{3} \frac{d^{3} y}{{d x}^{3}} + 2 x^{2} \frac{d^{2} y}{{d x}^{2}} + 2 y = 10 (x + \frac{1}{x})

Commented by infinityaction last updated on 01/May/22

l e t z = \log_{e} x \Rightarrow x = e^{z}

{D (D - 1) (D - 2) + 2 D (D - 1) + 2} y = 10 (e^{z} + e^{- z})

w h e r e \frac{d}{d z} = D

A uxiliary Equation

m^{3} - m^{2} + 2 = 0

(m + 1) (m^{2} - 2 m + 2) = 0

m = - 1, 1 \underline{+} i

C . F = c_{1} e^{- x} + e^{z} (c_{2} \cos z + c_{3} \sin z)

C . F = c_{1} x^{- 1} + x {c_{2} \cos (\log x) + c_{3} \sin (\log x)}

P . I = \frac{1}{(D + 1) (D^{2} - 2 D + 2)} 10 (e^{z} + e^{- z})

P . I = 10 {\frac{1}{(D + 1) (D^{2} - 2 D + 2)} e^{z} + \frac{1}{(D + 1) (D^{2} - 2 D + 2)} e^{- z}}

P . I = 10 {\frac{e^{z}}{2} + \frac{1}{D + 1} \cdot \frac{e^{- z}}{1 + 2 + 2}}

P . I = 10 {\frac{e^{z}}{2} + \frac{1}{5} \frac{1}{(D + 1)} e^{- z}}

P . I = 10 {\frac{e^{- z}}{2} + \frac{e^{- z}}{5} \frac{1}{(D - 1 + 1)} \cdot 1}

P . I = 10 {\frac{e^{z}}{2} + \frac{{z e}^{- z}}{5}} = 5 x + 2 x^{- 1} \log x

y = c_{1} x^{- 1} + x {c_{2} \cos (\log x) + c_{3} \sin (\log x)} + 5 x + 2 x^{- 1} \log x

Commented by cortano1 last updated on 28/Apr/22

n i c e