I could not figure this problem during lecture notes, please help me solve it! ANSWER ONLY THE QUESTION IN SQUARE. I set

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I could not figure this problem during lecture notes, please help me solve it! ANSWER ONLY THE QUESTION IN SQUARE. I set an example on the right if it helps. Thank you. Q: y'''+16y''=xsin4x
Q2: Solve the given Differential Equation by Undetermined Coefficient - Annil Approach. Annihilator y" +16y"=xsin4x Examele Solve the given differential equation by undetermined coefficients. y+4y= 4x+3eins-8 Solution The differential equation we can rewrite in the equivalent differential operater form (D²+4)y4c0ex+3-8 Step 1: First, we are the homogeneous equation y +4y=0 The auxiliary equation m² + 400 m²01 m=+21 complex conjugate roots Se, the complementary faction in Yeyear2x + eyrin2x Step 2: For the functions cors and sine we have a 0, P=1. andx-1=0 as, the operator D +1 annihilates the functions cors and sinx. For the function (-8) we have n-1=0 and the operater D sanihilates the fonction (-8). of these functions teorx+ 3eins-8 is sanihilated by the product of The operater D(D²+1). Now, sincs teorx+3rinx-8 is annihilated by the differential sperater & D(0+1), apply L₁= D(D² + 1) to the differential equation D(D+1)(D+4)y D(D+1)(4enes+leinx-8)=0 D(D+1)(D³+4)y=0 The auxiliary equation of the 5 order equation is mm² + 1)(²+4)=0 and has reett, -24 m₂ = 24 m₂ = -6 m₂ 6. mg 0. Thus, the generalinis yeyors + eins + eyes+in+ The first two terms for the complementary function of the original egestion. The remaining terms forma particular solution y, of the original equation % Acos + Brinx + C To find the specific coefficients A.B.C.and E abstitute y, sad i derivatives -Ain+Beass -Acosx-Binx in the original equation. y + 4y4ross +- ♥ -Acorx-Brinx+4(Acors Bains + C) = 4coxx+3ans- -Acors-Bains +44coss+48ins +40=4casx+3ine-8 -A+44 = 4 (A= 4/3 -8+48-3- 4C-8 Thus, the particular el i 8=1 (c=-2 y coss+sins-2 Step 3: The general solution of the original equation is y = y + yp y=c₁cos2s+c₂in2x+coss+sins-2