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The First-Order Linear Differential Equations - (2.3)
1. First-Order Linear Equations:
The general form of a linear differential equation:
dy
a 1 !x" ! 0
a 1 !x"
! a 0 !x"y " g!x",
dx
The standard form of a linear differential equation:
a !x"
g!x"
dy
P!x" " 0
, f!x" "
! P!x"y " f!x",
dx
a 1 !x"
a 1 !x"
A linear differential equation is homogenous if g!x" " 0 (or f!x" " 0), otherwise it is nonhomogeneous.
2. The Property:
The general solution of a first-order linear differential equation:
dy
! P!x"y " f!x" is of the form
dx
y " yc ! yp
where y c is the general solution of the homogeneous differential equation:
dy
! P!x"y " 0
dx
and y p is a particular solution of the nonhomogeneous differential equation:
dy
! P!x"y " f!x"
dx
Proof: Let y " y c ! y p . Then y # " y #c ! y #p , and
dy
! P!x"y " f!x" $ !y #c ! y #p " ! P!x"!y c ! y p " " !y #c ! P!x"y c " ! !y #p ! P!x"y p " " 0 ! f!x" " f!x"
dx
3. Method of Solving First-Order Differential Equations:
The above property suggests that the general solution of a first-order linear differential equation can be
obtained in two steps:
dy
! P!x"y " 0.
a. Find the general solution y c of
dx
dy
! P!x"y " f!x".
b. Find a solution y p of
dx
dy
Now let us derive the general solution for the equation
! P!x"y " f!x" by these two steps.
dx
dy
! P!x"y " 0 in a. is separable. The general
a. Observe that the homogeneous differential equation
dx
solution can be solved as follows.
1 dy " "P!x"dx $
y
#
1 dy "
y
# "P!x"dx,
ln|y| " " # P!x"dx ! C,
y c " Ce
b. Observe also that
d
dx
ye
# P!x"dx
"
dy
dx
e
# P!x"dx ! ye # P!x"dx d
"
dy
dx
e
# P!x"dx ! ye # P!x"dx P!x"
"e
Then
1
dx
# P!x"dx
# P!x"dx dy ! P!x"y " e # P!x"dx f!x".
dx
#
" P!x"dx
ye
# P!x"dx "
#
yp " e
The general solution of
d
dx
ye
#
" P!x"dx
# P!x"dx dx " e # P!x"dx f!x"dx
#
# e # P!x"dx f!x"dx
dy
! P!x"y " f!x" :
dx
y " y c ! y p " Ce
#
" P!x"dx
!e
#
" P!x"dx
# e # P!x"dx f!x"dx " e "# P!x"dx
C ! #e
# P!x"dx
# P!x"dx f!x"dx
The function e
is called the integrating factor of the differential equation.
Steps of Compute y :
a. Compute h!x" " # P!x"dx
the integrating factor: e h!x" .
b. Compute k!x" " # e h!x" f!x"dx.
c. Solution: y " e "h!x" !C ! k!x"".
dy
" 3y " 6 ! 2x " e x
dx
a. h!x" " # "3dx " "3x
Example Solve
the integrating factor is: e "3x
b. k!x" " # e "3x !6 ! 2x " e x "dx " " 20 e "3x " 2 xe "3x ! 1 e "2x
9
3
2
2
1 "2x
20
2
3x
"3x
"3x
e
xe
e
x ! 12 e x
c. y " e 3x C " 20
"
"
"
!
"
Ce
9
3
2
9
3
The general solution:
y " Ce 3x " 20 " 2 x ! 1 e x
2
9
3
Example Solve the initial value problem:
dy
! xy " x ! 1, y!4" " 1
dx
dy
! 2 x y " x2 ! 1 , for x ! %3
dx
x "9
x "9
!x 2 " 9"
x dx " 1 ln|x 2 " 9|
2
x2 " 9
1
2
the integrating factor is: e 2 ln x "9 " x 2 " 9
1
2
!x ! 1"
dx " # x ! 1 dx " !x 2 " 9" ! ln x ! !x 2 " 9"
b. k!x" " # e 2 ln x "9 2
x "9
x2 " 9
1
2
c. y " e " 2 ln x "9 C ! !x 2 " 9" ! ln x ! !x 2 " 9"
1
C ! !x 2 " 9" ! ln x ! !x 2 " 9"
"
x2 " 9
C
1
ln x ! !x 2 " 9"
"
!1!
2
2
x "9
x "9
d. y!4" " C ! 1 ! 1 ln 4 ! 7 " 1, C " " ln 4 ! 7
7
7
The solution of the initial value problem:
1
y"
" ln 4 ! 7 ! !x 2 " 9" ! ln x ! !x 2 " 9"
2
x "9
a. h!x" " #
2
Example Solve xy # ! !1 ! x"y " e "x sin!2x".
Write the equation in standard form:
y# !
1 ! x y " 1 e "x sin!2x"
x
x
a. h!x" " #! 1x ! 1"dx " ln|x| ! x
the integrating factor: e ln|x|!x " e ln|x| e x " xe x
b. k!x" " # e ln|x|!x 1x e "x sin!2x" dx " # xe x 1x e "x sin!2x"dx " # sin!2x"dx " " 12 cos!2x"
c. y " e "ln|x|"x C " 12 cos!2x" " 1x e "x C " 12 cos!2x"
The general solution:
y " 1x e "x C " 1 cos!2x"
2
Example Solve the initial value problem:
dy
! y " f!x" where f!x" "
dx
1, 0 $ x & 1
x, 1 $ x
a. h!x" " # dx " x
the integrating factor is e x .
b. k!x" " # e x f!x"dx "
c. y "
# e x dx " e x
if 0 $ x & 1
# xe x dx " xe x " e x
if 1 $ x
e "x !C 1 ! e x " " C 1 e "x ! 1
if 0 $ x & 1
e "x !C 2 ! xe x " e x " " C 2 e "x ! x " 1
if 1 $ x
d. y!0" " C 1 ! 1 " "1, C 1 " "2, for 0 $ x $ 1, y " "2e "x ! 1.
y!1" " "2e "1 ! 1, y!1" " C 2 e "1 ! 1 " 1 " C 2 e "1 " "2e "1 ! 1, C 2 " "2 ! e
The solution of the initial value problem:
y"
3
"2e "x ! 1
!"2 ! e"e
if 0 $ x & 1
"x
!x"1
if 1 $ x
, y!0" " "1.