1. LINEAR DIFFERENTIAL
EQUATION

2. Linear differential equation
 Definition
 Any function on multiplying by which the differential
equation M(x,y)dx+N(x,y)dy=0 becomes a differential
coefficient of some function of x and y is called an
Integrating factor of the differential equation.
 If μ [M(x,y)dx +N(x,y)dy]=0=d[f(x,y)] then μ is called
I.F

3.  Case-1
+P(x)y=Q(x)
 Is called a first order linear differential equation.
I.F=eP(x)dx
 The general solution is
Y(I.F)=

4.  Example-1
The equation is
Here P=1, Q=x
I.F=e1dx =ex
The solution is
Y(I.F)=
Yex= ex+c
=xex-ex+c
Y=x-1+ce-x

5.  Case-2
+P(y)x=Q(y)
 Is Linear differential equation
I.F=
 The general solution is
X(I.F)=

6.  Example-2
The equation is
Here P= , Q=
I.F=e-logy=
The solution is
X(I.F)=
X =

7. Bernoulli’s Equation
 Definition
 A differential equation is
 Case-1
The general solution is
yn
v(I.F)=

8. x2y6
y-5=x2
Put v=y-5
= -5y-6
+ x2
=-5x2
Here P= , Q=
I.F=e-5logx=x-5
The solution is
v(I.F)=
VX-5= dx +c
VX-5= dx +c
= x2 + c Y-5= x3 +cx5

9.  Case-2
The general solution is
v(I.F)=