FIRST ORDER DIFFERENTIAL EQUATIONS
FELIX HARVEY-ROSSER - 3457606
2D
1.
a)
dy
− y tan x = 2 sin x
dx
R(x) = e−
R
tan xdx
= e− ln sec x =
1
= cos x
sec x
d
(y cos x) = 2 sin x cos x
dx
Z
y cos x = sin 2xdx
y cos x =
1
cos 2x + c
2
1
y cos x = − (1 − 2 sin2 x) + c
2
1
y cos x = sin2 x − + c
2
y = sin x tan x + k sec x
———
b)
dy
+ y cot x = cos 3x
dx
R
R(x) = e cot xdx = eln sin x = sin x
d
(y sin x) = cos 3x sin x
dx
Z
y sin x = cos 3x sin x · dx
y sin x =
1
2
Z
sin 4x − sin 2x · dx
cos 4x cos 2x
−
+
+c
4
2
cos 4x cos 2x
y sin x = −
+
+c
8
4
1
y sin x =
2
1
2
FELIX HARVEY-ROSSER - 3457606
c)
1 − x2 y 0 − xy = 1
y0 −
R
R(x) = e
−x
dx
1−x2
x
1
y=
1 − x2
1 − x2
1
R
= e2
−2x
dx
1−x2
1
= e 2 ln(1−x
2)
=
p
1 − x2
p
0 √ 1 − x 2
2
=
y 1−x
1 − x2
Z
p
1
2
√
y 1−x =
1 − x2
p
y 1 − x2 = arcsin x + c
y=
arcsin x + c
√
1 − x2
———
d)
dy
− (1 + x)y = (1 − x2 )1/2
dx
√
dy
(1 + x)
1 − x2
−
y
=
dx 2(1 − x2 )
2(1 − x2 )
2(1 − x2 )
dy 1
1
1
−
y= √
dx 2 (1 − x)
2 1 − x2
R(x) = e
1
2
R
−1
dx
(1−x)
1
= e 2 ln(1−x)dx = (1 − x)1/2 =
√
√
1 1−x
d
y 1−x = √
dx
2 1 − x2
√
1
d
1
y 1−x = √
dx
2 1+x
Z
√
1
1
p
y 1−x=
dx
2
(1 + x)
√
√
y 1−x= 1+x+c
———
√
1−x
FIRST ORDER DIFFERENTIAL EQUATIONS
2.
a)
1
dy
+ y = x;
dx x
R
R(x) = e
1
dx
x
y(1) = 0
= eln x = x
d
(yx) = x2
dx
Z
yx = x2 dx
yx =
x3
+c=
3
x2
c
+
3
x
y(x) =
1
+c=0
3
1
c=−
3
y(1) =
y(x) =
x2
1
−
3
3x
———
b)
dy
= y + ex ;
dx
x = 0, y = 1
dy
− y = ex
dx
R(x) = e−
R
dx
= e−x
d
y · e−x = ex · e−x
dx
Z
−x
y · e = dx
y · e−x = x + c
y(x) = xex + cex
y(0) = c = 1
y(x) = xex + ex
3
4
FELIX HARVEY-ROSSER - 3457606
c)
dy
2
= xe−x − 2xy;
dx
x = 0,
dy
2
+ 2xy = xe−x
dx
R(x) = e
R
2xdx
= ex
2
d 2
2
2
y · ex = xe−x · ex
dx
d 2
y · ex = x
dx
x2
ye
Z
=
xdx
yex =
x2
+c
2
2
y(x) =
x2
2
+ c e−x
2
y(0) = c = 2
y(x) =
x2
2
+ 2 e−x
2
———
y=2
FIRST ORDER DIFFERENTIAL EQUATIONS
3.
m
dv
= mg − kv;
v(0) = 0
dt
k
dv
+ v=g
dt
m
k
let = ω02
m
dv
+ ω02 v = g
dt
R(t) = e
R
ω02 dt
2
= eω0 t
d 2
2
v · eω0 t = g · eω0 t
dt
v·e
ω02 t
Zt
=
2 0
g · eω0 t dt0
0
ω02 t
v·e
Zt
=g
2 0
eω0 t dt0
0
2 0
v·e
ω02 t
eω0 t
=g 2
ω0
2
t
0
e ω0 t − 1
v·e =g
ω02
g
2
v = 2 1 − e−ω0 t
ω0
kt
mg v(t) = ẋ(t) =
1 − e− m
k
ω02 t
———
5