Question

Solve i. and ii. Given the ordinary differential equation: cos(x)y' = sin(x)y + 1 Find the general solution of the given differential equation. ii. Solve the ordinary differential equation: ay' + by = a cos(wx) + Bsen(wx) Where: a, b, a,ß and w are nonzero real constants.

Answer 1

And i briven differential equation coses) y = Sim (2) 4 + 1 cos(x) y'-sin(x4 = 1 - y - Sime y Cosx Cosa y'- tarix y = Secla This is the first order differential equation So tan(x) abe I.F. then =é log(seclas) e log (seccass' sec(x) then solution. Y. (IF) Specia). (I.F.) dhe te s secas. I dete → y. Seclay Secla - x + c у Seclay x.sec (2) + C Sec(x) → y = then General salution y = x secca) + C secca)

Ans. i) Given the differential equation ay't by 2008 (wsl) +ß Sinius) where 2,ß, a, b. and w real constant non. Bero - y't by Coslu) + B Sin lose) a This is the first order linear differential equation then I.F. ehåbe box - e then solution Y (T.F.) = Se coswx) + £ sin wa)).exdx. + å J108011d2e **+ & sintwox) et de to yeux

let bla o cosuse e a I = ( cos(wa) ed ade of IZ = S Subraxe) et de then we will use formula Sunda= usudbe - (d . Sube detc I, = fost use) ebad de Sosin (wse we had a costwa). eat wa (simwx) e ad de = costwa) ell take [ simualada Srczowwe te he] box + wa2 sin (walean wa? Scos(wale at de a cos(wx) eta + wa² sin (wa) e ad_ wa? (I) b =ą costwa) e b 62 62 6² 62 b sinlose) e box 62 It wa2 I, = a coswalean + wa2 (1+ufe 2 JI + wa2 (Siniwa).exe) ban a collwale 62 b I = a .62 cos(wx). ebben + wa? 6² Sinlux). ebax (6²+ wa 2).6² (6²+wa2).b box => I = Siniwale ab 6² w22 cos(wa) bax+ wa2 6²+waz

bet de b a box was costwa) ed at de b a Sin(wsl) e a x a²w co8lwa) e bax Is = Isinux) e ax che bax Siniwal.es Scost oswn) we b a sin(wale a Sin wa) eta " + w?a? , sinwa). et a dx - a² w Iz = a siniwal é 042? (I.) - 2) Iz 62 Coswajebat 6² b 62 eban a²w cos(wa) e box + b 62 w2a2 62 b - Iz = sin(wx) e ax_ a²w.b? cos(wx) ex (6²_Q²w2).6 (6²a²we).6² a.62 >- ab 6²_a2w2 Sinewale dx a²w (62_a²wr) Cos(wx) e bax

Then bar X(I) + B (Iz) + c y.e > y = q (I). ex + B (Iz) e box ce box → y = x Londrina (03/wx) + wa2 sinwe); Elab Siniwx-a²w coswx] à (6²w2a² 6²_a²w2 + ceax > y = ab cos(wx)+xaw b²+w²22 b²tw²22 Sin/wx) + Bb Sinlws) - Baw Cos(wx)+ce and b²w²a2 6²-a²w2 then general salution bx paw + "ab b²+ a2w2 Law (6²+a²w2 Bb 62_a²w2 2) sin(wx)+ce 62-2?w2 )63(w)