Page 1
(1) The order of ODE: the order of the highest derivative
e.g.,
Chapter 14 First-order ordinary differential equation
order) (second order),(first 2
2
dx
yd
dx
dy
(2) The degree of ODE: After the equation has been rationalized, the power of
the highest-order derivative.
e.g.,
ODE degree second andorder third the
)( and )()(0)( 23
3
3
332/322/3
3
3
dx
yd
dx
yd
dx
dy
dx
dyyx
dx
dyx
dx
yd
(3) The general solution of ODE contains constants of integration, that may
be determined by the boundary condition.
(4) Particular solution: The general solution contains the constants which are
found by the boundary condition.
(5) Singular solution: Solutions contain no arbitrary constants and cannot be
found from the general solution.
Page 2
with n parameters satisfies an nth-order ODE in general. The boundary conditions on the solutions determine the parameters.
xaxay cossin 21
Chapter 14 First-order ordinary differential equation
14.1 General form of solution
),.....,,,,( 321 naaaaxfy
Ex: Consider the group of functions
equationorder -second 0
cossin
sincos
2
2
212
2
21
ydx
yd
xaxadx
yd
xaxadx
dy
Page 3
Chapter 14 First-order ordinary differential equation
14.2 First-degree first-order equation
0),(),(or ),( dyyxBdxyxAyxFdx
dy
Separable-variable equation
dxxfyg
dyygxf
dx
dy)(
)()()(
1)2
exp(
)2
exp()2
exp(1
2)1ln(
1
)1( :Ex
2
22
2
xAy
xAC
xy
Cx
yxdxy
dy
yxxyxdx
dy
Page 4
Chapter 14 First-order ordinary differential equation
Exact equations
dy
ydFdxyxA
yyxB
y
yxU
yFyFdxyxAyxU
cyxU
yxdUx
yxB
y
yxA
y
yxU
xx
yxU
y
y
yxUyxB
x
yxUyxA
dyy
yxUdx
x
yxUyxdUdyyxBdxyxA
yxU
)(]),([),(
),(
by determined be can )( and )(),(),(
ODE of solution the is ),( so,
0),(),(),(
)),(
()),(
(
),(),( and
),(),(
),(),(),(),(),(
satisfies which ),( function aFor
Page 5
Chapter 14 First-order ordinary differential equation
solution the is 2
3
2
3
)(0
)(2
3)()3(),(
exact is equation the1 ,1
),( ,3),(0)3(
03 :Ex
2
12
2
2
1
2
1
cxyx
ccyxx
cyFdy
dFx
dy
dFx
cyFyxx
cyFdxyxyxU
x
B
y
A
xyxByxyxAxdydxyx
yxdx
dyx
Page 6
Chapter 14 First-order ordinary differential equation
Inexact equations: integrating factors
)(1
)( })(exp{)()( if (b)
)(1
)( })(exp{)( )()(1
)(
)( if (a)
it find to solved be can )(or )( if (2)
it findingfor method general no ),( if (1)
)()(),(factor gintegratin an by gmultiplyin by
exact made be always can aldifferenti The
),(),(but 0),(),(
y
A
x
B
Aygdyygyy
x
B
y
A
Bxfdxxfxdxxfdx
x
B
y
A
B
d
dx
dB
x
B
y
A
dx
dB
x
B
y
A μ
x
yx
yx
Bx
Ay
yx
BdyAdx
x
yxB
y
yxAdyyxBdxyxA
Page 7
Chapter 14 First-order ordinary differential equation
cyxx
cyFyFyxyFyxy
U
cyFyxxyFdxyxxyxU
ydyxdxyxx
xxdxx
xx
yyxyx
B
y
A
Bxf
yx
By
y
A
xyyxByxyxAxydydxyx
x
y
ydx
dy
234
2'3'3
123422
322
2
22
is solution The
)(0)(2)(2
)()()34(),(
ODEexact an is 02)34(
)ln2exp(}1
2exp{)(2
)26(2
1)(
1)(
exact.not is ODE The 26
2),( ,34),(02)34(
2
32 :Ex
Page 8
Chapter 14 First-order ordinary differential equation
Linear equations
solution the is )()()(
1)()()(
)()(])([)(
)()()()((1) Eq.
})(exp{)()()()(
ODEexact an is 0))()()(()(
))()()(()(
)1()()()()()(
)(factor gintegratin multiply )()(
dxxQxx
ydxxQxyx
xQxyxdx
d
dx
xdy
dx
dyxyxPx
dx
dyx
dxxPxxPxdx
xd
dxxQyxPxdyx
xPxQxdx
dyx
xQxyxPxdx
dyx
xxQyxPdx
dy
Page 9
Chapter 14 First-order ordinary differential equation
)exp(2
)exp())(exp(2)2ln(
22
22
)2(2
method separated-Variable (2)
)exp(2
)exp(2)exp(4)exp(
)exp(}2exp{)( (1)
42 :Ex
2
222
2
222
2
xky
xkcxycxy
xdxy
dyxdx
y
dyyx
dx
dy
xcy
cxdxxxxy
xxdxx
xxydx
dy
Page 10
Chapter 14 First-order ordinary differential equation
Homogeneous equations
x
dx
vvF
dv
vvFdx
dvxvF
dx
dvxv
dx
dy
vxy
x,yfyxf
yxf
yxBxyyxA
yxByxA
x
yF
yxB
yxA
dx
dy
n
)(
)()(
onsubstituti the Making
)(),( obeys
it , anyfor If n. degree shomogeneou of function a is ),( (2)
degree. third the with and e.g., degree,
same the of functions shomogeneou ),( and ),( Where(1)
)(),(
),(
3322
Page 11
Chapter 14 First-order ordinary differential equation
)(sin)sin()ln()ln(sin
ln)ln(sinsin
cos
lncot
)tan(set
)tan( :Ex
1
12
1
AxxyAxx
yAx
x
y
cxcvdvv
v
cxx
dxvdv
vvdx
dvxv
dx
dyv
x
y
x
y
x
y
dx
dy
Page 12
Chapter 14 First-order ordinary differential equation
Isobaric equations
mvxydxxm
dyy
yxB
yxA
dx
dy
onsubstituti a make then , and to relative weight
a given each are and if consistent llydimensiona is equation The
),(
),(
cxxycxyxcxv
x
dxvdv
vxdx
dv
x
v
dx
dvvx
xx
v
vxdx
dvxvx
dx
dy
vdxxdvxdyvxym
yxdydxx
y
xy
yxdx
dy
ln2
1ln)(
2
1ln
2
1
122
)2
(2
1RHS
2
1LHS
2/2/1
lyrespective 1,2m 0, 1,2m is litydimensiona the 02)2
(
)2
(2
1 :Ex
222/12
21
2
2/12/12/3
2/32/12/1
2
2
Page 13
Chapter 14 First-order ordinary differential equation
Bernoulli’s equation
ODElinear )()1()()1(
)()1()()1(
)()()1
(
1)1(
linear nonlinear onsubstituti a make
1or 1 where )()(
11
1
xQnvxPndx
dvv
yxQnyxPn
dx
dv
v
yxQyxP
dx
dv
n
y
dx
dv
n
y
dx
dy
dx
dyyn
dx
dv
yv
nnyxQyxPdx
dy
nn
n
nn
n
n
Page 14
343
33333
13
3
33
33
434
44341
43
6 is solution the
6)6(1
)1
()()()(
1)(
1}
3exp{})(exp{)(..
6)( ,3
)( with ODElinear 63
23
12
3
33let
2 :Ex
cxxy
ycxxdxxxx
dxxQxx
xv
xdx
xdxxPxFI
xxQx
xPxvxdx
dv
xx
y
dx
dvyx
x
y
dx
dvy
dx
dvy
dx
dy
dx
dyy
dx
dvyyv
yxx
y
dx
dy
Chapter 14 First-order ordinary differential equation
Page 15
Miscellaneous equations
Chapter 14 First-order ordinary differential equation
)(
onsubstituti a make
)( (1)
xbFadx
dyba
dx
dv
cbyaxv
cbyaxFdx
dy
11
11
2
2
2
)1(tan
tan1
111
)1( :Ex
cxyx
cxvdxv
dv
vdx
dy
dx
dvyxv
yxdx
dy
Page 16
ODE shomogeneou a
0)((
0)()(
shomogeneou is RHS and let (2)
fYeX
bYaX
dX
dY
gfefYeXgYfX e
cbabYaXcYbXa
YyXxgfyex
cbyax
dx
dy
Chapter 14 First-order ordinary differential equation
2223
123
1
2
2
)32)(34()21
1)(1
1
14()1(
)3exp()2)(14()2ln(3
2)14ln(
3
1ln
23
2
143
4
472
42
42
472
42
52
42
52let
42
52
10642 and 0352
,let 642
352 :Ex
cxyxycx
y
x
yx
cvvXcvvX
dX
dX
v
dv
v
dvdv
vv
v
v
vv
dX
dvX
v
v
vXX
vXX
dX
dvXv
dx
dvXv
dX
dYvXY
YX
YX
dX
dY
YyXxyx
yx
dx
dy
Page 17
14.3 Higher-degree first-order equation
Chapter 14 First-order ordinary differential equation
0),().....,(),(),( is solution general The
,...2,1for ),( equation of solution the is 0),(
),( and ),(
0))........()((
for 0),(),(...........),(
321
21
011
1
yxGyxGyxGyxG
niyxFdx
dypyxG
yxFpyxFF
FpFpFpdx
dypyxapyxapyxap
n
ii
iii
n
nn
n
0)]1()][1([ is solution general The
0)1()1ln(ln1
202)1( (2)
0)1(1lnln1
0)1( (1)
0]2)1][()1[(
for 02)123()1( :Ex
221
222
22
2
11
2
22223
xkyxky
xkycxyx
xdx
y
dyxy
dx
dyx
xkyc)(xyx
dx
y
dyy
dx
dyx
xypxypx
dx
dypxyypxxpxxx
Page 18
dy
dp
p
F
y
F
pdy
dx
dx
dyppyFx
x
1
for ),(
for soluable Equation
Chapter 14 First-order ordinary differential equation
xyyxy
pxypyxpypy
ypyp
kkxyykxkyy
kx
y
ky
y
kpkpyc
ypcyp
dyyp
dpp
dy
dpy
dy
dpyp
dy
dpypypyp
dy
dpy
dy
dp
p
y
ppdy
dxpy
p
yx
dxdypyxppy
p
solutionsingular 038)6/1(94
9)16()3()6( equation. origional the Change
6/1061(2)
solution general 6303603)(6
1lnlnln2ln
2202 (1)
0)2)(61(12613
363
/for 036 :Ex
2322
22222222
22
23322
22
2
22
2
222
2
22
Page 19
Chapter 14 First-order ordinary differential equation
dx
dp
p
F
x
Fp
dx
dypxFy
y
),(
for soluable Equation
solutionsingular 0021 (2)
solution general 4)(
4)2()(eq. origion the intoput
1lnln
202 (1)
0)2)(1(
0)1(2)1(0)1(2
2222
02 :Ex
2
222
2
2
22
2
yxyxxp
kxky
kxxpykkxp
cx
px
dx
p
dp
dx
dpxp
dx
dpxpp
pdx
dpxpppp
dx
dpxp
dx
dpxp
dx
dpxppp
dx
dyxpxpy
yxpxp
Page 20
Chapter 14 First-order ordinary differential equation
Clairaut’s equation
ODE origional the in eliminate 0),(0 (2)
solution general )()(
)(eq. origional the intoput
0 (1)
0)(
1112
12211
212
2
ppxGxdp
dF
cFxcycFc
cFxccxccdx
dyp
cxcydx
yd
dx
dp
xdp
dF
dx
dp
dx
dp
dp
dF
dx
dpxpp
dx
dy
)( pFpxy
solutionsingular 04442
2
02 (2)
solution general ))(()( (1)
0)2(2
:Ex
2222
22
2
yxx
yxx
yx
ppx
ccxcFxyppF
pxdx
dpp
dx
dpp
dx
dpxp
dx
dy
ppxy