Question

Solve the following initial value problem

dy dc 2 +-y = ln(2), y(1) = 0.

Answer 1

Quei- Solve the following initial value problem dy + 2 y = logo Y(1) = 0 da 2 Salh dy + 2 y = loga (۱) da 2 Which is linear differential equation in 'y' and compairing with dy + P(ac) y = then doi Q (20. P(x) = 2 and Q(8) = loga x Now, Integrating factor Sp da ga ada 21 ta 20 e. I.F. = - е. е. = 2loga loga? е, - I-F:= = x² The general solution of (1) is given by you x (I.F.) I.F.) x Q (C) dal + c where "Ć constant of integration :

y x x² = fazloga da to 3 11 loga.no - a sloga fa da te by integration of two functions af loga - 5* 23 dor + c posloga - {Sadol + c Yxx² = x loga pas long ae - $ x on + c 3 la to on 1 ats - - y xx² = x logge - 1 Dividing by 'n? both sides yra= I loga -ta -(2) using Y(1)=0 Putting x=1, In (2) 0 =$(1) - į log 1 - 5x1 + $ 22 O 含X xo to to c=1 T 9

Putting the value of 'c' in (3), we get Y CON = & loga 1 x a of Ans 90²