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

19.4 First-order linear equations: use of an integrating factor 557
and the steady-state terms are
[
ωsinωt +
VRC
R 2 C 2 ω 2 +1
cos ωt
]
RC
EXERCISES19.4.4
1 Find the general solution ofthe followingequations:
dy
(a)
dx +y=1 (b) dy
dx +2y=6
(c)
(e)
dx
dt +6x=4
dy
dx =6y+9
(d) dy
dx −3y=2
(f) dx
dt =3x−8
2 Find the particular solution ofthe following
equations:
(a) dy
dx +4y=7, y(0)=1
(b) dx
dt −x=4,
(c) dy
dt =3y+2,
(d) dy
dx =4y−8,
x(0)=2
y(0)=2
y(1)=2
3 Find the general solution of dx
dt =2x+4t.Whatis
the particular solution which satisfiesx(1) = 2?
4 Find the general solution of dy
dx +y=2x+5.
5 Solve dx
dt =t−tx,x(0)=0.
6 Useanintegratingfactortoobtainthegeneralsolution
ofiR+L di = sin ωt,whereR,Land ω areconstants.
dt
7 Solvex dy
dx +y=x4 .
8 Use an integrating factorto findthe general solution
oft dx
dt +x=3t.
9 Find the general solution of dx +2xt =t.Findthe
dt
particular solution satisfyingthe condition
x(0) = −1.
10 Find the general solution of
tẋ+3x= et
t 2
Solutions
1 (a) y=1+ce −x
(b) y=3+ce −2x
(c) x= 2 3 +ce−6t
(d) y=ce 3x − 2 3
(e) y=ce 6x − 3 2
(f) x= 8 3 +ce3t
2 (a)y= 7 4 − 3 4 e−4x
(b)x=6e t −4
(c)y= 8 3 e3t − 2 3
(d)y=2
3 −2t−1+ce 2t ,−2t−1+5e 2(t−1)
4 2x+3+ce −x
5 1−e −t2 /2
6
7
8
9
L((R/L)sinωt − ωcosωt)
R 2 +L 2 ω 2
x 4
5 + c x
3t
2 + c t
1
2 − 3 2 e−t2
10 et +c
t 3
+ce −Rt/L