Question

Solve the following 1st order ODE: * + 5x = cos(2t) x(0) = 2

Answer 1

① Given ist order one is de +52 Set re(o) = 2 form ap eq Initial condition is o in drogerelor DX +52 = coszt (:D ) Dt 5) = Coszt the the solution to ca o is in rect) = def + 385 foron of - where GF is characteriske fonction Rp. I is particular ontegral &c.E the root of can be determined by using aunialary function from to is in the form of f(0) = 2CH) eq GIO) Dsch on compare get) a casat Aurialary foncton foto wit 5-0 ma-5 .. ecif(t) = comt iXcF CH) = ce

sep.a(t) = Coszt Rationalise with Dog' ne (075) coszt 0-5) 6+5) 6. mapa (t) e (mat) - @th) (2-6) Sinat 2- . Dlarat) 5 calzt - 2 fins 5-25 Coral ps)- s() = -2 Sint 6-28 -25 we know that - Sinat if H-&)*0 Sinat fo) it fra) to -2 Sina Scoile (t) - -2. Simt 4-25 56-S24 -- impa (t) = 2 sinat t 1.- - - substitute eq ②, er in eq . . xH) = cest + 2 sinzt & he cost o C: Given enitial condition no=2 weets substitute in es ☺ (0) =2= cet q sino te caso 1 Sinozo Color)

carosa ca substitute this 'c' value in ea & Garesal siti solution is sta sinzt + e coszt