This document discusses techniques for solving different types of first-order differential equations. It focuses on solving nonlinear, non-separable equations of the form M(x,y)+N(x,y)y'=0. To do so, one must find an exact differential equation by determining if the partial derivatives are equal, making the equation exact. If the equation is not exact, an integrating factor can be used to make it exact. Once the equation is exact, the general solution can be found by computing M(x,y)dx + N(x,y)dy and integrating both sides. An example is provided to demonstrate reducing a non-exact equation to an exact one in order to find the general solution.

1. EXACT
EQUATIONS
Non-linear, Non-seperable first order differential equation

2. Techniques we know:
2
Differential
equation
Type
Solving
method
M(y)y'=n(x)
Non-linear
seperable
Seperation
of variables
y'+P(x)y=Q(x) Linear
Integrating
factor
M(x,y)+N(x,y)y'=0
Non-
seperable
non-linear
?

3. IMPLICIT DIFFERENTIATION
Let us take an example function:
f(x,y)=
=>f(x,y)'=( )'
=
Now differentiate the function partially:
By comaring above equations we get:

4. 4
So ,this is the general rule:
=M(x,y)+N(x,y)y'
=>To solve problems of the form:
M(x,y)+N(x,y)y'=0
What we do?
We should find: f(x,y)

5. Solving nonlinear non-
separable equations
• To find f(x,y):

6. EXAMPLE:
• Let the differential equation be :
•
• =>
commonpart
• &
• Common part=
• h(x)=
• g(y)= =>

7. If we try to find the general solution of
3y+2xy'=0
where
if we do
but f(x,y)=h(x)+commonpart+g(y)
but the above integrations doesn't
have commonpart or h(x) and g(y).
To find G.S for the above equation we
reduce it to exact equation.
let us know what is exact equation..

8. Let's know what is exact equation ?
• Remember , order doesn't matter for partial derivatives:
=>
•
• =>
• &
• =>
•
• =>
To solve the differential of the form M(x,y)+N(x,y)y'=0 the necessary
condition is
• => and the above equation is called exact.

9. To solve problems of the form
M(x,y)+N(x,y)y'=0
Step 1:- Observe
Step 2:-Compute Mdx + Ndy
Step 3:- write the general solution as
f(x,y)= M(x,y)dx+ N(x,y)dy
If a differential equation Mdx+Ndy=0
which is not exact can be made
exact by multiplying it with a
suitable µ(x,y)≠0 then µ(x,y) is called
an integrating factor of the equation
Mdx+Ndy=0.
9

10. Differential
equation
Type Method of
solving
M(y)y'=n(x) Non-linear
separable
Seperation of
variables
y'+P(x)y=Q(x) Linear Integrating
factor
M(x,y)+N(x,y)y'=0
Non-linear
Non separable
Exact
Method of
exact
equations
SUMMARY