334 Chapter 11First-order differential equations
11.8 Exercises
Section 11.2
State the order of the differential equation and verify that the given function is a solution:
- y 1 = 1 A 1 cos 12 x 1 + 1 B 1 sin 12 x
- y 1 = 12 x
3
1 + 13 x
2
1 + 14 x 1 + 15 4.
Find the general solution of the differential equation:
9.A body moves along the xdirection under the influence of the forceF(t) 1 = 1 cos 12 πt, where
tis the time. (i)Write down the equation of motion. (ii)Find the solution that satisfies
the initial conditionsx(0) 1 = 10 andH(0) 1 = 11.
Verify that the given function is a solution of the differential equation, and determine the
particular solution for the given initial condition:
- y 1 = 1 cx
2
; y 1 = 1 24 whenx 1 = 12
- ; y 1 = 1 2 whenx 1 = 12
- y 1 = 1 ce
− 2 x1 − 1 1; y(0) 1 = 14
Section 11.3
Find the general solution of the differential equation:
Solve the initial value problems:
- y(0) 1 = 11 20. y(0) 1 = 1 − 1
- y(2) 1 = 11 22. y(0) 1 = 10
Solve the initial value problems:
- y(1) 1 = 12 24. y(1) 1 = 10
- y(2) 1 = 10
26.Show that a differential equation of the form
is reduced to separable form by means of the substitutionu 1 = 1 ax 1 + 1 by.
dy
dx
=++fax by c()
xy
dy
dx
xy
344
=+;
2
22
xy
dy
dx
=−()xy+ ;
dy
dx
xy
x
=
;
dy
dx
e
xy=;
+
dy
dx
yy
xx
=
−
;
()
()
1
1
dy
dx
x
y
=
−
;
2
1
21
dy
dx
y
x
=
−
;
2
3
dy
dx
y
x
=
dy
dx
y =−yy( 1 )
dy
dx
e
2 x=
dy
dx
= 3 xy
2
dy
dx
= 4 xy
2
dy
dx
x
y
=
3
2
dy
dx
++=; 220 y
yce
x=
−
2
dydx
+=; 20 xy
x
dy
dx
=; 2 y
dy
dx
x
3
3
= 24
dy
dx
x
2
2
=cos 3
dy
dx
e
x=
− 3
dy
dx
=x
2
dy
dx
y
x
xy
xc
x
+=; =+
3
3
2
2
3
3
dy
dx
3
3
=; 12
dy
dx
y
2
2
+=; 40
dy
dx
yye
x−=; = − 22 1