Question

please explain, thank you

Solve the following differential equations for an unknown function f(t): (a) df(t) +2f(t) = X(0,2](t) (b) Sketch the solution for f(t) for 0 <t< 4. dt

Answer 1

+ 2 f(t) = x sol":- Given diffenential equation is d fet) (t) dt [0, 2 where Х is de fined [o, 2] [o 2 Then the given differential con te defined equation. dfit +2.fet) for tt [o, 2 X, (t=4 9,6 192] . d f(t) dt + 2 ft for t & [0, 1], Now the auxiliary equation of the of the given equation is +2 Then the plementary function is ca A é 2t particular solution Now when t€ [0,2], the of the given equation is (1) D +2 dt Ź 11+2) (1) 2 (1 + 2)7.(1) = { (i-+-) 2 P.I- 2 Hence the solution of the given equation is fet) = A ezt + Ź for to [0,2

when * & [0, 2 fct) = A e 2t Hence the solution of the solution of the given equation can. com tire form written m -zt fet) - A e (+) + + X[0, 2 21 (AW) 6 Now when te [0,2 When t EC2, 4 ft) f(t) an to Then graph of fet) is following :- 2 (Aus)