Question

a. Find a particular solution to the nonhomogeneous differential equation y" + 16y = cos(4x) + sin(4x). Yo = (xsin(4x))/8-(xcos(4x))/8 help (formulas) b. Find the most general solution to the associated homogeneous differential equation. Use ci and C2 in your answer to denote arbitrary constants. Enter c1 as c1 and C2 as c2. Un = c1cos(4x)+c2sin(4x) help (formulas) c. Find the solution to the original nonhomogeneous differential equation satisfying the initial conditions y(0) = 3 and y'(0) = 2. y = help (formulas)

Answer 1

& solution - Given tial de equation Y"+16 Y = COS 148 ) + sin 149 ) particular homogenous soletion of of the non the non IQ 192 differential equation is yp = pid where pol = 02 +46 and @la) und Qlay = 00344251048 Yp= [(05 (4a) + sinlya)] D2H6 - sio 42 - 02+)6 Cos14a ) + I 02716 cos(400 t sinua = - -42 +16 - -42716 == cos(4a) + á sinux = case failure case 2 ease up = o(a) yp : " [10842 +510427 2b

Yo te the li cosya sinua de SinVX Sinu con de general Solution to the homogener de fferential equations Auxililiary equation of the given diferenti differential equation is m2 +16=0 matul the solution 18 x = arcos4 a +c sinya (C). Required Solution is y = Yc + YP y = ci cosya +(2 Sin4a + sinya - COS 49 - 0 8 8 given initial condition are Y10) -- 3, 4'10)=2 Now Y10) = 4, COS 460) + ( sin 410)+Sin410) -cos460) 8 4 CICU +42009 +0 09= 3+ 33 = 2558

You :-4 C Sin 48 +03 144 +46 (0542 + 4Cosy 74511 ? 4. 45in 49 Y'co) = -49 Sie 35-49 Sin 410)+46 COS410) + (0541043) 08470), 310410) 2. 4C160) +46210 +=2 "462 = 2-4 462 = 3/2 C2 = 3/8 substituting these values in equation o ya) = {cos 4x + 3 sin 48 + sinxa - rosua 2403401 + sinya lyca ) = 3 C0848 + L sinyal is a requived solution