Overview
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-
Here
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-
Provided
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-
Sol.
Here,
So that-
Thus the equation is exact and its solution is-
Which means-
Or
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-
Sol.
We can write the given equation as-
Here,
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-
Sol.
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-
Sol.
Here given,
M = 2y and N = 2x log x – xy
Then-
Here
Then
Now multiplying equation (1) by 1/x, we get-
4. For the following type of equation-
An I.F. is
Where-
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