Question

O Mo Tu We Th Fr Sa Su Memo No. Date Question 4 Consider the inhomogeneous clifferential equation Ä + 5x+6X = Cost - 3 Sint and the associated homogeneous differential equation X+5x+6x=0 @ Formulate the characteristic equation and solve it 6 Hence write down the general Solution of the homogeneous differential equations @ Determine values of a and b such that Xple) = a cost +6 Sint is a particular Solution of the inhomogeneous differential equaliten @ Hence write down an expression for the Solution of the inhomogeneous differential que @ Determine the solution of the inhomogeneous differental equation that satischer X (0) = 1,

Answer 1

ü tsit6u- cost-3 sint The characterstic equation will m2 +5m + 6=0 mu +3m + 2m + 620 ) m(m+3) +2 (MP3) - =) (m13) Im12) " me-2, -3 homoginiony dobtenential of genenal solution erueton will be le = cië 2-27 + q e 3t a cost & bsint - asint + 6 cost hp się - a cost + (-6 sint) put this of a cost absint +51-asint +6Cost) t6( a cost tosint) a cost -sint -a +56 + 6a 1 -b osat 66= -3 56+ 5a=1 56-5a = 3 65 ) a: 2 .. lxp- cost + (1/3) sint 2020105121 1

@ geneny solubers will be U- Uct xp =>/x-a ezt thest + 2 cost - sint X(0)=1 i 200=0 - G+4+2 -1 -24 -362-f30 9=2 G=-715 by pont @ x= 26 24 - 7 e3t + 2 cost - sint

I use characterstic equation to solve this problem