(D^3 −2D^2 +9D−18)y=6cos3x


Question and Answers Forum

All Questions Topic List
Differential Equation Questions
Previous in All Question Next in All Question
Previous in Differential Equation Next in Differential Equation
Question Number 56580 by subhankar10 last updated on 18/Mar/19

$(D^{3} - {2 D}^{2} + 9 D - 18) y = 6 \cos 3 x$
	Answered by tanmay.chaudhury50@gmail.com last updated on 19/Mar/19
	$let y=e^(mx) be a solution Dy=me^(mx) D^2 y=m^2 e^(mx) .. so complimentary function m^3 e^(mx) −2m^2 e^(mx) +9me^(mx) −18e^(mx) =0 e^(mx) (m^3 −2m^2 +9m−18)=0 e^(mx) ≠0 m^3 −2m^2 +9m−18=0 m^2 (n−2)+9(m−2)=0 (m−2)(m^2 +9)=0 m=2 and ±3i C.F C_1 e^(2x) +C_2 e^(i3x) +C_3 e^(−i3x) Particular intregal y=((6cos3x)/(D^3 −2D^2 +9D−18)) =6×((cos3x)/(D^2 (D−2)+9(D−2))) =6×((cos3x)/((D−2)(D^2 +9))) =6×(((D+2) cos3x)/((D^2 −4)(D^2 +9))) =(6/((−3^2 −4)))×((−3sin3x+2cos3x)/(D^2 +9)) =((−6)/(13))×((−3sin3x+2cos3x)/(D^2 +9)) =(6/(13))×(((3sin3x−2cos3x))/(D^2 +9)) =(6/(13))×((rsin(3x−θ))/(D^2 +9)) [r=(√(3^2 +2^2 )) =(√(13)) tanθ=(2/3)] p=(6/((√(13)) ))×((cos(3x−θ))/(D^2 +9)) q=(6/(√(13)))×((sin(3x−θ))/(D^2 +9)) p+iq=(6/(√(13)))×(e^(i(3x−θ)) /(D^2 +9)) p+iq=(6/(√(13)))×e^(−iθ) ×(e^(i×3x) /(D^2 +9)) p+iq=(6/(√(13)))×e^(−iθ) ×(e^(i3x) /((D+i3)(D−i3))) p+iq=(6/(√(13)))×(cosθ−isinθ)×(e^(i3x) /((i3+i3)(D+i3−i3)))×1 =(6/(√(13)))×((3/(√(13)))−i×(2/(√(13))))×(((cos3x+isin3x))/(6i))×x =(1/(13))×((3i+2)/i^2 )×(xcos3x+ixsin3x) =((−1)/(13))(i3xcos3x−3xsin3x+2xcos3x+i2xsin3x) =((−1)/(13)){(2xcos3x−3xsin3x)+i(3xcos3x+2xsin3x)} =((−x)/(13))(2cos3x−3sin3x)+((−ix)/(13))(3cos3x+2sin3x) now our answer is related to complex part so answer is Particular intregal is =((−x)/(13))(3cos3x+2sin3x) so complte answer is=C.F+P.I y=C_1 e^(2x) +C_2 e^(i3x) +C_3 e^(−i3x) +((−x)/(13))(3cos3x+2sin3x)$
	$l e t y = e^{m x} b e a s o l u t i o n$ $D y = {m e}^{m x}$ $D^{2} y = m^{2} e^{m x}$ $. .$ $s o$ $c o m p l i m e n t a r y f u n c t i o n$ $m^{3} e^{m x} - 2 m^{2} e^{m x} + 9 {m e}^{m x} - 18 e^{m x} = 0$ $e^{m x} (m^{3} - 2 m^{2} + 9 m - 18) = 0$ $e^{m x} \neq 0$ $m^{3} - 2 m^{2} + 9 m - 18 = 0$ $m^{2} (n - 2) + 9 (m - 2) = 0$ $(m - 2) (m^{2} + 9) = 0$ $m = 2 a n d \pm 3 i$ $C . F$ $C_{1} e^{2 x} + C_{2} e^{i 3 x} + C_{3} e^{- i 3 x}$ $P a r t i c u l a r i n t r e g a l$ $y = \frac{6 c o s 3 x}{D^{3} - 2 D^{2} + 9 D - 18}$ $= 6 \times \frac{c o s 3 x}{D^{2} (D - 2) + 9 (D - 2)}$ $= 6 \times \frac{c o s 3 x}{(D - 2) (D^{2} + 9)}$ $= 6 \times \frac{(D + 2) c o s 3 x}{(D^{2} - 4) (D^{2} + 9)}$ $= \frac{6}{(- 3^{2} - 4)} \times \frac{- 3 s i n 3 x + 2 c o s 3 x}{D^{2} + 9}$ $= \frac{- 6}{13} \times \frac{- 3 s i n 3 x + 2 c o s 3 x}{D^{2} + 9}$ $= \frac{6}{13} \times \frac{(3 s i n 3 x - 2 c o s 3 x)}{D^{2} + 9}$ $= \frac{6}{13} \times \frac{r s i n (3 x - θ)}{D^{2} + 9} [r = \sqrt{3^{2} + 2^{2}} = \sqrt{13} t a n θ = \frac{2}{3}]$ $p = \frac{6}{\sqrt{13}} \times \frac{c o s (3 x - θ)}{D^{2} + 9}$ $q = \frac{6}{\sqrt{13}} \times \frac{s i n (3 x - θ)}{D^{2} + 9}$ $p + i q = \frac{6}{\sqrt{13}} \times \frac{e^{i (3 x - θ)}}{D^{2} + 9}$ $p + i q = \frac{6}{\sqrt{13}} \times e^{- i θ} \times \frac{e^{i \times 3 x}}{D^{2} + 9}$ $p + i q = \frac{6}{\sqrt{13}} \times e^{- i θ} \times \frac{e^{i 3 x}}{(D + i 3) (D - i 3)}$ $p + i q = \frac{6}{\sqrt{13}} \times (c o s θ - i s i n θ) \times \frac{e^{i 3 x}}{(i 3 + i 3) (D + i 3 - i 3)} \times 1$ $= \frac{6}{\sqrt{13}} \times (\frac{3}{\sqrt{13}} - i \times \frac{2}{\sqrt{13}}) \times \frac{(c o s 3 x + i s i n 3 x)}{6 i} \times x$ $= \frac{1}{13} \times \frac{3 i + 2}{i^{2}} \times (x c o s 3 x + i x s i n 3 x)$ $= \frac{- 1}{13} (i 3 x c o s 3 x - 3 x s i n 3 x + 2 x c o s 3 x + i 2 x s i n 3 x)$ $= \frac{- 1}{13} {(2 x c o s 3 x - 3 x s i n 3 x) + i (3 x c o s 3 x + 2 x s i n 3 x)}$ $= \frac{- x}{13} (2 c o s 3 x - 3 s i n 3 x) + \frac{- i x}{13} (3 c o s 3 x + 2 s i n 3 x)$ $n o w o u r a n s w e r i s r e l a t e d t o c o m p l e x p a r t$ $s o a n s w e r i s$ $P a r t i c u l a r i n t r e g a l i s$ $= \frac{- x}{13} (3 c o s 3 x + 2 s i n 3 x)$ $s o c o m p l t e a n s w e r i s = C . F + P . I$ $y = C_{1} e^{2 x} + C_{2} e^{i 3 x} + C_{3} e^{- i 3 x} + \frac{- x}{13} (3 c o s 3 x + 2 s i n 3 x)$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum