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

626 Chapter 20 Ordinary differential equations II
REVIEWEXERCISES20
1 UseEuler’smethodwithh = 0.1 to estimatex(0.4)
given dx
dt =x2 −2xt,x(0) =2.
2 Thegeneralization ofEuler’smethodto the two
coupledequations
dy 1
dx = f(x,y 1 ,y 2 )
dy 2
dx =g(x,y 1 ,y 2 )
is given by
y 1(i+1)
= y 1(i)
+hf(x i ,y 1(i)
,y 2(i)
)
y 2(i+1)
= y 2(i)
+hg(x i ,y 1(i)
,y 2(i)
)
Given the coupled equations
dy 1
dx =xy 1 +y dy 2
2
dx =xy 2 +y 1
estimatey 1 (0.3)andy 2 (0.3),ify 1 (0) = 1 and
y 2 (0) = −1.Takeh = 0.1.
3 Expressthe followingequations asasetoffirst-order
equations:
(a)
d 2 y
dt 2 +6dy dt +8y=0
(b) 3 d2 x
dt 2 +5dx dt +4x=0
(c) 4 d3 x
dt 3 +8d2 x
dt 2 +6dx dt +5x=0
4 Expressthe followingcoupledfirst-order equationsas
asingle second-order differentialequation:
(a)
(b)
(c)
dy 1
dx =3y 1 +4y 2 ,2dy 2
dx = 6y 1 +8y 2
dx 1
dt
dy 1
dt
=6x 1 −5x 2 ,2 dx 2
dt
=8y 1 +4y 2 ,2 dy 2
dt
= 4x 1 −3x 2
= 4y 1 −6y 2
Solutions
1 4.7937
4 (a)
2 0.7535, −0.7535
dy
3 (a) 1 dy (b)
=y 2
dt 2 = −8y
dt 1 −6y 2
dx
(b) 1
=x
dt 2 3 dx 2
(c)
= −4x
dt 1 −5x 2
dx
(c) 1 dx
=x 2
dt 2 =x
dt 3
4 dx 3
= −5x
dt 1 −6x 2 −8x 3
d 2 y
dx 2 −7dy dx = 0
d 2 x
dt 2 − 9 dx
2 dt +x=0
d 2 y
dt 2 −5dy dt −32y=0