This document has everything you need to pass your A1 for engineering mathematics 214, every topic from, NAMELY, EXACT EQUATIONS, HOMOGENOUS DE, BERNOULLI'S EQUATION, REDUCTION OF ORFER, UNDETERMINED COEFFICIENTS, VARIATION OF PARAMETERS, . The file is concise, organized, color-coded and very neat....
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)
(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)
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