Solve the following differential equations :
(i) If a differential equation is ,
then y(I.F) = ∫Q.(I.F)dx + c, where I.F = e∫Pdx
(ii) ∫dx = x + c
Given:-
This is a linear differential equation, comparing it with
P = 1, Q = cosx
I.F = e∫Pdx
= e∫dx
= ex
Solution of the equation is given by
y(I.F) = ∫Q.(I.F)dx + c1
⇒ y ex = ∫cosx. ex dx + c1
let I = ∫ ex cosxdx
= cosx∫ exdx ∫(sinx∫exdx)dx + c2
using integrating by part
I = ex cosx + ∫sinxexdx + c
= ex cosx [sinx∫exdx∫(cosx∫exdx)dx] + c2
⇒ I = ex cosx + sinxex–I + C2
⇒ 2I = (cosx + sinx)ex + C2
putting I