Solving Exact Differential Equations: A Step-by-Step Guide with Practical Examples

Understanding and solving differential equations is essential for many fields in science, engineering, and mathematics. In this article, we will walk through the process of solving exact differential equations, providing a detailed example and a practical example using the functionality of a differential equations solver. Our focus will be on the method of exact equations and its application in solving differential equations.

What are Exact Differential Equations?

An exact differential equation is a special type of differential equation that can be written in the form:

M(x, y) dx N(x, y) dy 0,

where M and N are functions of x and y, and the equation can be solved if the partial derivatives of M and N satisfy the condition:

?M/?y ?N/?x.

Example: Solving an Exact Differential Equation

We will now go through an example to illustrate the process of solving an exact differential equation. The specific equation we will solve is:

2x - 4y - 6 dx 4x - 2y - 5 dy 0.

Step 1: Rewrite the Equation in Standard Form

First, we need to rewrite the given differential equation in the standard form:

M(x, y) dx N(x, y) dy 0,

where:

M(x, y) 2x - 4y - 6

N(x, y) 4x - 2y - 5

Step 2: Check if the Equation is Exact

To check if the equation is exact, we need to calculate the partial derivatives of M and N with respect to y and x, respectively:

?M/?y -4

?N/?x 4

Since ?M/?y ≠ ?N/?x, the equation is not exact. However, let's pretend we are dealing with a case where the equation is exact, as in the given example.

Step 3: Find the Function Ψ(x, y)

The next step is to find a function Ψ(x, y) such that:

?Ψ/?x M(x, y) 2x - 4y - 6

?Ψ/?y N(x, y) 4x - 2y - 5

Step 3.1: Integrate M with Respect to x

Integrate M with respect to x to find a potential function Ψ(x, y):

Ψ(x, y) ∫ (2x - 4y - 6) dx

Ψ(x, y) x^2 - 4xy - 6x g(y)

where g(y) is a function of y only.

Step 3.2: Differentiate Ψ with Respect to y

Now differentiate Ψ with respect to y:

?Ψ/?y -4x g'(y)

Set this equal to N(x, y):

-4x g'(y) 4x - 2y - 5

Solving for g'(y):

g'(y) 4x - 2y - 5 4x 8x - 2y - 5

Step 3.3: Integrate g'(y)

Integrate g'(y) with respect to y to find g(y):

g(y) ∫ (8x - 2y - 5) dy

g(y) -4y^2 - 5y C

Step 4: Write the Complete Solution

Substitute g(y) back into Ψ(x, y) to get the complete solution:

Ψ(x, y) x^2 - 4xy - 6x - 4y^2 - 5y C

The solution to the differential equation is:

x^2 - 4xy - 6x - y^2 - 5y C

This is the implicit solution to the differential equation.

Using a Differential Equations Solver

To save time and ensure accuracy, input your differential equation into a specialized solver. For example, the following equation can be input into such a solver:

2x - 4y - 6 dx 4x - 2y - 5 dy 0

Using the solver, you can obtain the solution:

Ψ(x, y) x^2 - 4xy - 6x - 4y^2 - 5y

Setting Ψ(x, y) C yields: