Answer to Question #189362 in Differential Equations for Farid Bahar

Question

(d^2 y)/〖dx〗^2 +2 dy/dx+5y=34cos2x

Answer 1

Given equation is-

$\dfrac{d^2y}{dx^2}+2\dfrac{dy}{dx}+5y=34cos2x$

Its Auxilary equation is-

$m^2+2m+5=0\\[9pt] m=\dfrac{-2\pm\sqrt{4-20}}{2}=\dfrac{-2\pm4i}{2}=-1\pm 2i$

The roots are- $m= -1+2i \text{ and } -1-2i$

Then complimentary function is-

$C.F. = e^{-x}(c_1cos2x+c_2sin2x)$

Particular Integral-

$PI=\dfrac{43cos2x}{D^2+2D+5}$

$=\dfrac{34cos2x}{-4+2D+5}\\[9pt]=\dfrac{34cos2x}{2D+1}\times \dfrac{2D-1}{2D-1}\\[9pt]=\dfrac{34(2D-1)cos2x}{4D^2-1}\\[9pt]=\dfrac{34(-4sin2x-cos2x)}{4(-4)-1}\\[9pt]=8sin2x+2cos2x$

Hence The complete solution is-

y= CF+PI

$y=e^{-x}(c_1cos2x+c_2sin2x)+8sin2x+2cos2x$