Let's begin by understanding what differential equations are. A differential equation is an equation involving one or more terms and the derivatives of one dependent variable with respect to the other independent variable. An exact differential equation is a specific type of ordinary differential equation commonly utilized in physics and engineering. Exact differential equations are widely used in thermodynamics for calculating internal energy and also used in electrostatics for calculating electric potential.
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In this article, we will cover the Exact differential equations. This concept falls under the broader category of differential equations. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of five questions have been asked on this concept, including one in 2019, one in 2020, two in 2021, and one in 2022.
A differential equation is an equation involving one or more terms and the derivatives of one dependent variable with respect to the other independent variable.
Differential equation: dy/dx = f(x)
Where “x” is an independent variable and “y” is a dependent variable
Example of differential equation: $x \frac{d y}{d x}+2 y=0$
The above-written equation involves variables as well as the derivative of the dependent variable $\mathrm{y}$ with respect to the independent variable $\mathrm{x}$. Therefore, it is a differential equation.
The following relations are some of the examples of differential equations:
(i) $\frac{d y}{d x}=\sin 2 x+\cos x$
(ii) $\mathrm{k} \frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dx}^2}=\left[1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^2\right]^{3 / 2}$
Exact Differential Equation
The equation A (x,y) dx + B (x,y) dy=0 is an exact differential equation if there exists a function of two variables x and y having continuous partial derivatives such that the exact differential equation definition is separated as follows
ux(x, y) = A(x, y) and uy (x, y) = B(x, y);
General Form
The general form of the exact differential equation is where A and B are the polynomial functions in terms of x and y.
Sometimes some differential equations can be solved using observation only. In such equations, we can get a differential of a function of x and y.
If the differential equation A (x, y) dx + B (x, y) dy = 0 is not exact, it is possible to make it exact by multiplying using a relevant factor u(x, y) which is known as integrating factor for the given differential equation.
Consider an example,
2ydx + x dy = 0
Now check it whether the given differential equation is exact using testing for exactness.
The given differential equation is not exact.
In order to convert it into the exact differential equation, multiply by the integrating factor u(x,y)= x, the differential equation becomes,
2 xy dx + x2 dy = 0
The above resultant equation is an exact differential equation because the left side of the equation is a total differential of x2y.
Sometimes it is difficult to find the integrating factor. But, there are two classes of differential equations whose integrating factor may be found easily. Those equations have the integrating factor having the functions of either x alone or y alone.
When you consider the differential equation A (x,y) dx + B (x,y) dy=0, the two cases involved are:
Case 1: If [1/B(x,y)][Ay(x, y) – Bx(x,y)] = h(x), which is a function of x alone, then e∫h(x)dx is an integrating factor.
Case 2: If [1/A(x,y)][Bx(x, y) – Ay(x,y)] = k(y), which is a function of y alone, then e∫k(y)dy is an integrating factor.
The following steps explain how to solve the exact differential equation:
Step 1: The first step to solving the exact differential equation is to make sure the given differential equation is exact.
Step 2: Write the system of two differential equations that defines the function u(x,y). That is
Step 3: Integrating the first equation over the variable x, we get
Step 4: Differentiating concerning y, substitute the function u(x,y) in the second equation
Step 5: We can find the function φ (y) by integrating the last expression so that the function u(x,y) becomes
Step 6: Finally, the general solution of the exact differential equation is given by
u (x,y) = C.
Illustration 1 :
Let us first separate terms containing only x with dx and terms containing only y with dy
Here first two terms have both x and y. We can make an observation that first two terms are the differentiation of x2y. Hence we can write this equation as
Integrating this, we get
This is the solution of this equation
The presence of the following exact differentials should be observed in a given differential equation
Illustration 2 :
Observe that
So the equation is
Integrating
Example 1: The solution of the differential equation dy/dx = -1 is
Solution:
As we have learned
The general form of Variable Separation -
Hence, the answer is x+y = C.
Example 2: The solution of a differential equation is
Solution:
As we have learned
The general form of Variable Separation -
is also a linear differential equation that we can write
Hence, the answer is .
Example 3: Solution of differential equation
Solution:
As we have learned
The general form of Variable Separation -
The equation can be written as
Hence, the answer is .
Example 4: The solution of the differential equation is
Solution:
As we have learned
The general form of Variable Separation -
The equation can be written as
Hence, the answer is
Example 5: The solution of the differential equation is
Solution:
As we have learned
The general form of Variable Separation -
The equation can be written as
dividing both sides by we get
on integrating, we get
Hence, the answer is .
Exact differential equations are a powerful tool in mathematics and physics, providing a method for finding potential functions from differential relationships among them. These equations are expanded in various applications and fields, including thermodynamics, electrostatics, and fluid mechanics, where they help to describe fundamental physical processes with the help of integrable functions.
It describes the rate of change in quantity and is used in science, engineering, business, etc. It can model many phenomena in different fields.
An exact differential equation is a type of first-order differential equation that is solved with the help of integration.
Exact differential equations are generally first order.
Exact differential equations are generally solved by observations and after this, we integrate them.
Exact differential equations are widely used in thermodynamics, electrostatics, and fluid mechanics.
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