Question

Q.8 Find the Particular integral solution
of the D.E (D2 - 4D + 3)y = 20 cos x*
O a) yp = 4 cosx - 3 sinx
O b) yp = 2 sinx - 4 cosx
Oc) yp = - 3 cosx + 4 sinx
O d) yp = 2 cosx - 4 sinx

Answer 1

$\textbf{Given:}$

$\textsf{Differential equation is}$

$\mathsf{(D^2-4D+3)y=20\;cosx}$

$\textbf{To find:}$

$\textsf{The particular integral of the given D.E}$

$\textbf{Solution:}$

$\underline{\textsf{Particular integral:}}$

$\mathsf{=\dfrac{20\;cosx}{D^2-4D+3}}$

$\mathsf{=\dfrac{20\;cosx}{-1-4D+3}}\;\;\mathsf{\;\;(D^2\implies\;-1)}$

$\mathsf{=\dfrac{20\;cosx}{-4D+2}}$

$\mathsf{=\dfrac{20\;cosx}{-2(2D-1)}}$

$\mathsf{=\dfrac{-10\,cosx}{2D-1}}$

$\mathsf{=\dfrac{-10\,cosx}{2D-1}{\times}\dfrac{2D+1}{2D+1}}$

$\mathsf{=\dfrac{-10\,cosx(2\,D+1)}{4D^2-1}}$

$\mathsf{=\dfrac{-20\,D(cosx)-10\,cosx}{4(-1)-1}\;\;(D^2\implies\;-1)}$

$\mathsf{=\dfrac{-20\,D(cosx)-10\,cosx}{-4-1}}$

$\mathsf{=\dfrac{-20(-sinx)-10\,cosx}{-5}}$

$\mathsf{=-4\,sinx+2\,cosx}$

$\mathsf{=2\,cosx-4\,sinx}$

$\implies\boxed{\mathsf{P.I=2\,cosx-4\,sinx}}$

$\textbf{Answer:}$

$\textbf{Option (d) is correct}$

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Answer 2

Step-by-step explanation:

yp = 2 cosx - 4 sinx

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