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What is an exact differential equation?

by Harpreet_Physics


An exact differential equation is formed by differentiating its solution directly without any other process,

Is called an exact differential equation if it satisfies the following condition-


is the differential co-efficient of M with respect to y keeping x constant and

is the differential co-efficient of N with respect to x keeping y constant.

Step by step method to solve an exact differential equation

1. Integrate M w.r.t. x keeping y constant.

2. Integrate with respect to y, those terms of N which do not contain x.

3. Add the above two results as below-


Example-3: Determine whether the differential function ydx –xdy = 0 is exact or not.

Solution.  Here the equation is the form of  M(x , y)dx + N(x , y)dy = 0

           But, we will check for exactness,

These are not equal results, so we can say that the given diff. eq. is not exact.

Example: Solve-



So that- 

Thus the equation is exact and its solution is-

Which means-


Equation reducible to the exact form

1. If M dx + N dy = 0 be an homogenous equation in x and y, then 1/ (Mx + Ny) is an integrating factor (M dx + N dy 

Example: Solve-


We can write the given equation as-


Multiply equation (1) by 1/x^4 we get-

This is an exact differential equation-

2. I.F. for an equation of the type

IF the equation Mdx + Ndy = 0 be this form, then 1/(Mx – Ny) is an integrating factor.

 Example: Solve-


Here we have-

Now divide by xy, we get-

Which is an exact differential equation-

3. In the equation M dx + N dy = 0,

(i) If  be a function of x only = f(x), then  is an integrating factor.

(ii) If be a function of y only = F(x), then is an integrating factor.

Example: Solve-


Here given,

M = 2y and N = 2x log x – xy




Now multiplying equation (1) by 1/x, we get-

4. For the following type of equation-

An I.F. is


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