3. EXACT DIFFERENTIAL EQUATION
We see that these differential equations can be obtained
by directy diffrentiating their solutions. Differential
equations of this type are called exact equations and bear
the following property:
An exact differential equation can always be obtained
from its primitive directly by differentiation, without any
subsequent multiplication, elimination etc..
4. Working Rule
If the equation M dx+N dy=0 satisfies the condition
𝝏M/ 𝝏𝒚 = 𝝏N/𝝏x then it is exact. To integrate it,
Integrate M with regard to x regarding 𝒚 as constant;
Find out those terms in N which are free from x and
integrate them with regard to 𝒚;
Add the two expressions so obtained and equate sum
to an arbitraty constant.
5. .
NON EXACT DIFFERENTIAL EQUATION
• For the differential equation
𝑀(𝑥, 𝑦) 𝑑𝑥 + 𝑁(𝑥, 𝑦)𝑑𝑦 = 0
IF 𝝏𝑴/𝝏𝒚≠𝝏𝑵/𝝏𝒙 then,
𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒕𝒊𝒂𝒍 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝒊𝒔 𝒔𝒂𝒊𝒅 𝒕𝒐 𝒃𝒆 𝑵𝑶𝑵𝑬𝑿𝑨𝑪𝑻
•If the given differential equation is not exact then make
that equation exact by finding INTEGRATING FACTOR.
6.
Methods to find an INTEGRATING FACTOR (I.F.) for given
non exact equation:
M(x, y)dx + N(x, y)dy = 0
Rule 1: If 1/𝑁(𝜕𝑀/𝜕𝑦+𝜕𝑁/𝜕𝑥) f𝑥 {i.e. function of x only}
Then I.F. = e∫ 𝒇(𝒙)𝒅x
Rule 2: If 1/-𝑀(𝜕M/𝜕𝑥−𝜕N/𝜕𝑦)=f(y) only a function of y,
Then I.F. = e∫ 𝒇(𝑦 )𝑑𝑦
7. Rule 3:If the given differential equation is
homogeneous with 𝑀𝑥 + 𝑁𝑦 ≠0
Then I.F. =1/𝑀𝑥+𝑁y
Rule 4: If the equition is of the form
yf(x,y)dx + xg(x,y)dy =0
Then I.F.= 1/Mx-Ny
