find-particular-solution-of-D-3-4D-y-sin-2x-by-using-inverse-operation-

Question Number 118574 by bramlexs22 last updated on 18/Oct/20

f i n d p a r t i c u l a r s o l u t i o n o f

(D^{3} + 4 D) y = \sin 2 x b y u s i n g

i n v e r s e o p e r a t i o n ?

Answered by TANMAY PANACEA last updated on 18/Oct/20

y = \frac{s i n 2 x}{D (D^{2} + 4)}

y = \frac{e^{i 2 x}}{D (D^{2} + 4)}

= \frac{e^{i 2 x}}{(i 2) {{(D + i 2)}^{2} + 4}}

= \frac{e^{i 2 x}}{(i 2) (D^{2} + 4 i D)}

= \frac{e^{i 2 x}}{(i 2)} \times \frac{1}{D \times 4 i \times (1 + \frac{D}{4 i})}

= \frac{1}{- 8} \times \frac{1}{(1 + \frac{i 2}{4 i})} \times \frac{e^{i 2 x}}{1} \times x

= \frac{1}{- 8} \times \frac{2}{3} \times \frac{c o s 2 x + i s i n 2 x}{1} \times x

= \frac{x}{- 12} (c o s 2 x) + i \times \frac{x}{- 12} (s i n 2 x)

P . I = \frac{x}{- 12} \times s i n 2 x \to y = \frac{x s i n 2 x}{- 12}

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