, Equations
2
.
4
-
If f is a 2-variable function , the differential off is :
at
of ax yo
=
+
af = Mdx + NdY
-
A DE (MdX + NdY = 0) is called exact if (Max + Nay) is exact
(Max + Nay is exact if and only
if N
-
-
Note : If (Max + Nay = 0) is exact with /x = M and /Y = N
, then :
df = Max + Nay = 0 => f(x , y) =
Examples
t DE : Cocoax -
sinsinay =
isa constant
(i) (Cocosy) dy + (-sinsing) dy Let M = COSXCOSY (iii) Define f = /Max
-
M N N = -
S1x sing f =
S (cocosy)dX
f = SinX(OSY + 9(Y)
(ii) verify if exact
(iv)
M =2 (COSX(OSY) of
Then
=
-
cosiny
o
ON(sinsing) &) Sinxcosy
= -
Cosysiny
a + g(y)) = -
sinsing o
M EN
exact sing) + g'(y)
sinsing
:
= = -
-
g'(y) = 0
: 9 (y) = < (constant)
Consequentially , if f = sinx cosy then af = Max + Nay = 0 , so that f(x , y) = Sincosy = C
,Examples
-
#2
Solve the DE : (2 x + y) ax + (x + 6y) dy = 0
(iv) Then
(i)((x + y)dX
Of No
+ (x + 6y) = 0 Let M = 2 x + y
-
x
MN N =
X + 6Y YX + (y) = X+
is constant
ciic Verify if exact DE (iii) Define f =
/Max 0 +x+ g'(y) = *+ by
M 2(2x + y) =
= 0+1 = 1 =
((2x + y)dx 9'(y) = Gy
by Gy
=
2x2
-
+ yx + g(y) 9(y) =
(by dy
_
2X
(X + 6) = H =
= x2 + yX + 9(y) =
342 + k
&M EN
=
:: exact De
=
consequentially , if f = x+ xy + 3y2 is such that df = Max + Ndy
Since , af = Max + Ndy = 0 it follows that f(x , y) = * + X Y + 34 = (CER)
,
examples
X 6x/y xy 3/n/y)
As :
-
+ + y + xy -
3/4 + 3
x)ay
(1 -
+ y=
is constant
(i) (1 -
3/y + x) qy + y= 3/x -
1 cis verify if exact (iii) Define f = /m
dX
M =(1-3/
+) =
((1 -
3/x + y)dx
(1 -
3/4 + x) dy = (3/x -
1 -
y)dx =
X -
3/n/x1 + xy + 9(y)
(1-3/ )
=
(1 -
3/y + x) dy -
(3/X -
1 -
y)dX = 0
(iv)
(1 3/x + y)dX + (1 3/y + x) dy
N
-
-
= 0 Then
-- -
M C
M N .: exacte T
=
f(x 9(y)) 1
3
-
3(n(x) + xy + =
-
+ x
-
24
Let M= 1 -
3/x + y
N= 1 -
3/y + X
*+ g'(y) = 1 -
3/y +*
g'(y) = 1 -
3/y
consequentially , if f = x -
3///y + y -
3/nly) is such that df = Max + Nay g(y) =
((1 -
3/y) dy
Since af Max + Ndy = 0 X 3/n/x / + xy + y 31/y) y
= -
3/(y) + k
it follows that f(x , y)
-
-
= =
,
,
= C (CER)
, Examples
A (54-2x)
a
: -
( = 0
(i) (5y (ii) Verify if exact Smax
2x)a (iii) Define f
-
-
24 = 0 =
(( 24)dx
My 2) (y)
-
=
=
-
=
-
2
Y
(5y 2x) dy
- = (2y)dX =
-
2 x y + 9(Y)
-
(5y 2x)dy-
+ ( (y)dX = 0
( 2y)dX
-
+ (54 -
2x)dY = 0 (iv) Then F = N
- -
M N
Y
6) -
2 x y + g(y)) = 5y 2x
-
by
Let M = -
24
N =
5y -
2x -x+ 9/(y) = 54 -X
g'(y) =
54
consequentially , if f =
-2xy + 54c is such that df = Max + Nay g(y) ((5y)ay
=
Since 5)
by
af Max + Ndy = 0 2 xY +
it follows that f(x , y)
-
, = = =
,
= C (CER)
Examples as (exty(ax + (2 + x + yex) dy =
Y(0) = 1
(i) (ex + y)dx yeY) dy
f
+ (2 + x+ = 0 (ii) verify if exact (iv)Then N
=
- -
M N
M G(ex + y)
= = 1
by Gy
o
(e
+ Xy + 9(y)) = 2 + X + Yey
x
xtye
N
Let M =
e +y = L+
N = 2 +x+ yeY *+ g'(y) = 2 + *+ yeY
9(y) Ity or
M EN : exact De =
=
(iii) Define f = Smax g(y) =
((2 + yeY)dy
=
((ex + y)dx =
((dy Sye dy
+
=
ex + xy + 9(y) = (y + (yey eY) + k
-
consequentially , if f = ex + xY + 24 + yeY-eY is such that df = Max + Nay
Since , af = Max + Ndy = 0
, it follows that f(x , y) = ex + xy + 2y + yes ey
-
= C (CER)
2
.
4
-
If f is a 2-variable function , the differential off is :
at
of ax yo
=
+
af = Mdx + NdY
-
A DE (MdX + NdY = 0) is called exact if (Max + Nay) is exact
(Max + Nay is exact if and only
if N
-
-
Note : If (Max + Nay = 0) is exact with /x = M and /Y = N
, then :
df = Max + Nay = 0 => f(x , y) =
Examples
t DE : Cocoax -
sinsinay =
isa constant
(i) (Cocosy) dy + (-sinsing) dy Let M = COSXCOSY (iii) Define f = /Max
-
M N N = -
S1x sing f =
S (cocosy)dX
f = SinX(OSY + 9(Y)
(ii) verify if exact
(iv)
M =2 (COSX(OSY) of
Then
=
-
cosiny
o
ON(sinsing) &) Sinxcosy
= -
Cosysiny
a + g(y)) = -
sinsing o
M EN
exact sing) + g'(y)
sinsing
:
= = -
-
g'(y) = 0
: 9 (y) = < (constant)
Consequentially , if f = sinx cosy then af = Max + Nay = 0 , so that f(x , y) = Sincosy = C
,Examples
-
#2
Solve the DE : (2 x + y) ax + (x + 6y) dy = 0
(iv) Then
(i)((x + y)dX
Of No
+ (x + 6y) = 0 Let M = 2 x + y
-
x
MN N =
X + 6Y YX + (y) = X+
is constant
ciic Verify if exact DE (iii) Define f =
/Max 0 +x+ g'(y) = *+ by
M 2(2x + y) =
= 0+1 = 1 =
((2x + y)dx 9'(y) = Gy
by Gy
=
2x2
-
+ yx + g(y) 9(y) =
(by dy
_
2X
(X + 6) = H =
= x2 + yX + 9(y) =
342 + k
&M EN
=
:: exact De
=
consequentially , if f = x+ xy + 3y2 is such that df = Max + Ndy
Since , af = Max + Ndy = 0 it follows that f(x , y) = * + X Y + 34 = (CER)
,
examples
X 6x/y xy 3/n/y)
As :
-
+ + y + xy -
3/4 + 3
x)ay
(1 -
+ y=
is constant
(i) (1 -
3/y + x) qy + y= 3/x -
1 cis verify if exact (iii) Define f = /m
dX
M =(1-3/
+) =
((1 -
3/x + y)dx
(1 -
3/4 + x) dy = (3/x -
1 -
y)dx =
X -
3/n/x1 + xy + 9(y)
(1-3/ )
=
(1 -
3/y + x) dy -
(3/X -
1 -
y)dX = 0
(iv)
(1 3/x + y)dX + (1 3/y + x) dy
N
-
-
= 0 Then
-- -
M C
M N .: exacte T
=
f(x 9(y)) 1
3
-
3(n(x) + xy + =
-
+ x
-
24
Let M= 1 -
3/x + y
N= 1 -
3/y + X
*+ g'(y) = 1 -
3/y +*
g'(y) = 1 -
3/y
consequentially , if f = x -
3///y + y -
3/nly) is such that df = Max + Nay g(y) =
((1 -
3/y) dy
Since af Max + Ndy = 0 X 3/n/x / + xy + y 31/y) y
= -
3/(y) + k
it follows that f(x , y)
-
-
= =
,
,
= C (CER)
, Examples
A (54-2x)
a
: -
( = 0
(i) (5y (ii) Verify if exact Smax
2x)a (iii) Define f
-
-
24 = 0 =
(( 24)dx
My 2) (y)
-
=
=
-
=
-
2
Y
(5y 2x) dy
- = (2y)dX =
-
2 x y + 9(Y)
-
(5y 2x)dy-
+ ( (y)dX = 0
( 2y)dX
-
+ (54 -
2x)dY = 0 (iv) Then F = N
- -
M N
Y
6) -
2 x y + g(y)) = 5y 2x
-
by
Let M = -
24
N =
5y -
2x -x+ 9/(y) = 54 -X
g'(y) =
54
consequentially , if f =
-2xy + 54c is such that df = Max + Nay g(y) ((5y)ay
=
Since 5)
by
af Max + Ndy = 0 2 xY +
it follows that f(x , y)
-
, = = =
,
= C (CER)
Examples as (exty(ax + (2 + x + yex) dy =
Y(0) = 1
(i) (ex + y)dx yeY) dy
f
+ (2 + x+ = 0 (ii) verify if exact (iv)Then N
=
- -
M N
M G(ex + y)
= = 1
by Gy
o
(e
+ Xy + 9(y)) = 2 + X + Yey
x
xtye
N
Let M =
e +y = L+
N = 2 +x+ yeY *+ g'(y) = 2 + *+ yeY
9(y) Ity or
M EN : exact De =
=
(iii) Define f = Smax g(y) =
((2 + yeY)dy
=
((ex + y)dx =
((dy Sye dy
+
=
ex + xy + 9(y) = (y + (yey eY) + k
-
consequentially , if f = ex + xY + 24 + yeY-eY is such that df = Max + Nay
Since , af = Max + Ndy = 0
, it follows that f(x , y) = ex + xy + 2y + yes ey
-
= C (CER)
