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

546 Chapter 19 Ordinary differential equations I
dy
(k)
dx = 6sinx dx
(l)
y dt = 9cos4t
dy
x 2
(b)
dx = −ky, k constant
dx
(m)
dt = 3cos2t+8sin4t
dy
(c)
x 2 dx =y2
+x
(d) y dy
2 Find the particular solution ofthe following
dx =sinx
equations:
(e) y dy
dx
dx =x+2
(a) =3t, x(0) =1
dt (f) x 2dy
dy
(b)
dx = 6x2
dx =2y2 +yx
y , y(0)=1
dx
(g)
dy
(c)
dt = 3sint
dt = t4
x 5
, y(0)=2
y
5 Findthe general solutionsofthe followingequations:
dy
(d)
dx = e−x
y , y(0)=3
dx
(a)
dt =xt (b) dy
dx = x y
dx
(e)
dt = 4sint+6cos2t , x(0)=2 (c) t dx
x
dt =tanx
3 Find the general solution of dx
dx
= lnt.Find the (d)
dt dt = x2 −1
t
particular solution satisfyingx(1) = 1.
6 Findthe general solution ofthe equation
4 Find the general solutionsofthe following equations: dx
=t(x −2).Findthe particular solution which
dy
(a)
dx =kx, k constant dt
satisfiesx(0) = 5.
Solutions
1 (a) y=3x+c (b) x=5t+c 3 tln|t|−t+c,tln|t|−t+2
(c) y=x 2 +c (d) y=3t 2 +c
(e) y= 8 3 x3 +c (f) x= 3 4 (a) kx2
2 +c
4 t4 +c
(b)Ae −kx
y 2
(g)
2 = x3
3 +c (h) x 3
3 = t4 4 +c (c) − 1
x +c
x 2 y 3 (i)
2 =et e−2x
+c (j) =c − (d)y 2 =2(c −cosx)
3 2
y 2
(k)
2 =c−6cosx (l) x 3
3 = 9 (e)y 2 =x 2 +4x+c
4 sin4t+c x
(f)
x 3
(m)
3 + x2
2 = 3 A−2ln|x|
2 sin2t−2cos4t+c
2 (a) x= 3 (g) x6
2 t2 +1 (b) y 2 =4x 3 6 = t5 5 +c
+1
5 (a) x=Ae t2 /2 (b) y 2 =x 2 +c
y 2
(c)
2 =5−3cost (d) y2 =11−2e −x
(c) x = sin −1 (kt) (d) x = 1+At2
x 2
1−At 2
(e)
2 =3sin2t−4cost+6 6 2+Ae t2 /2 ,2+3e t2 /2