**Because U,H,A and G are all state functions**, their differentials are exact. We can derive a few relationships just from this fact. This equation tells you that the change in entropy in a system can be calculated by integrating (∂V∂T)P data.

dU, dG, dH etc are all exact differentials and the variables themselves are known as state functions because **they only depend on the state of the system**. However, dq and dw for example, are inexact differentials.

Approximate number is defined as a number approximated to the exact number and **there is always a difference between the exact and approximate numbers**. For example, are exact numbers as they do not need any approximation. But, , are approximate numbers as they cannot be expressed exactly by a finite digits.

A first-order differential equation (of one variable) is called exact, or an exact differential, **if it is the result of a simple differentiation**. The equation P(x, y)y′ + Q(x, y) = 0, or in the equivalent alternate notation P(x, y)dy + Q(x, y)dx = 0, is exact if P_{x}(x, y) = Q_{y}(x, y).

**An exact differential is sometimes also called a total differential, or a full differential, or, in the study of differential geometry, it is termed an exact form**. The integral of an exact differential over any integral path is path-independent, and this fact is used to identity state functions in thermodynamics.

**Solution:**

- Separate the variables and integrate both sides,
- We obtained the value of integrating factor, multiply this with our original equation. (x+xy
^{2})dx + x^{2}ydy=0. Checking for exactness again, - M(x, y)= Substitute to determine f(y)
- It follows f(y)=C where, C is a Constant. Therefore, the general solution becomes,

Determine whether the following differential is exact or inexact. **If it is exact, determine z=z(x,y).** **If this equality holds, the differential is exact**. Therefore, dz=(2x+y)dx+(x+y)dy is the total differential of z=x2+xy+y2/2+c.

**Because U,H,A and G are all state functions**, their differentials are exact. We can derive a few relationships just from this fact. This equation tells you that the change in entropy in a system can be calculated by integrating (∂V∂T)P data.

Exact Differential Equation Examples

Some of the examples of the exact differential equations are as follows : **( 2xy – 3x ^{2} ) dx + ( x^{2} – 2y ) dy = 0**. ( xy

**ϕdQ=0 −** not differential. Was this answer helpful?

Therefore, internal energy is a state function (i.e. exact differential), while **heat and work are path functions (i.e. inexact differentials)** because integration must account for the path taken.

In mathematics, some problems can be solved analytically and numerically. An analytical solution involves framing the problem in a well-understood form and calculating the exact solution. A numerical solution means **making guesses at the solution and testing whether the problem is solved well enough to stop**.

Definition:The differential equation M(x,y) dx + N(x,y) dy = 0 is said to be an exact differential equation **if there exits a function u of x and y such that M dx + N dy = du**.

Dated : 13-Jul-2022

Category : Education