Question

find th solution of differential equation

Find the solution & + ax = of the diffrorial equahon in terms of (An anbiny A sin Wt 410) ab 2

Answer 1

Given differential equation is 3 + ax = A sin we x(0)=b .. => + Q x = 'A so wiet which is in the form : de + PCO )= Q(6). So, here . ple) =a , alo) a Asin wit. a Therefore, integrating facks is - spat jade at. ze ze Now, multiplying Inf. in (1) , we get et dx tapat a te at IF aut sin at NA .- 2a - Apolo sin up » Xell - A cat sinue

Now, shear son we de a Aseal sin wat de no serwis Selos io- Jsema manus : (Using by poorts) Sin wt Cor wt e sin wt 02 Q2 ( again applying 4 by parts) Me sin web lucas \ Q? Iyoz de) The integral je al sin ut at appears again on the right side of the equation. . :-) A seat son wat die + A will lead sinust af Ar elesin unt weat cos with a - L a

= A a? +62), le sunt de 02 ," . Af det sin wt-we cos wit 7 A Selp son wat de sch uit de 1. Alaeaesin unt - we al corant) w? ta? - to Aae at sin wit Awe at (as well - " Netc = Weta? - ?ta? . co2to2 Ninw, A oeul san wt Aw cos wit 240² «112 - 1с ... ». X (H) = Aw AQ sin will W?ta² Coswt at - + c wator

Now given initial condition is x (o)=b. 1 . Aw + c = b : w2tQ² => . c = bt Aw : b(W2+Q²) + Aw - w²ta2 Weyaz Therefore, the unique Jeiesol solution is Aa son wit x ()= Aw Cos wit w2tQ² W2+Q² I a Ib(w2t0²) + Aw I - w2ta2 (Ans )