082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

and so, upon integrating,
so that
(cosx)y=x+c
y =
19.4 First-order linear equations: use of an integrating factor 549
x
cosx + c
cosx
EXERCISES19.4.1
1 Eachofthe followingequations is exact.Solvethem.
(a) x 2dy
dx +2xy=x3
(b)
1 dy
x 2 dx − 2 x 3y=5x3
(c) e x (
y + dy
dx
)
=cosx
Solutions
1 (a) y= x2
4 + C x 2 (b) y= 5x6
4 +Cx2 (c) y=e −x sinx+Ce −x
19.4.2 Apreliminaryresultinvolvingseparationofvariables
Consider the following differential equation forthe dependentvariable µ:
dµ
dx
= µP (19.3)
wherePissome function ofxonly. Usingseparation ofvariables wehave
1 dµ
µ dx =P
and integrating both sides
∫ 1
µ dµ = ∫
Pdx
∫
lnµ= Pdx
µ = e ∫ Pdx
Inthisdevelopmenttheconstantofintegrationhasbeenomitted.Thereasonforthiswill
be apparent in what follows. So the solution of Equation (19.3), for any functionP(x),
isµ=e ∫ Pdx .
For example, if P(x) = 1 dµ
, then Equation (19.3) is
x dx = µ and its solution is
x
µ=e ∫ (1/x)dx = e lnx =x.