Using an integrating factor

Let us consider a differential equation of the form.

where P ( x,y ) and Q ( x,y ) are functions of two variables x and y , continuous in some domain D . If

then the equation will not be a total differential equation. However, we can try to select a so-called integrating factor, which is a function µ (x, y), such that, after multiplying by it, the differential equation is transformed into a total differential equation. In this case, the following equality holds:

This condition can be written as:

The last expression is a first-order partial differential equation that determines the integrating factor µ ( x,y ).

Unfortunately, there is no general method for finding the integrating factor. However, we can mention some special cases for which the resulting partial differential equation can be solved and, as a result, the integrating factor determined.

In this case we have , so the equation for µ ( x,y ) can be written as:

The right-hand side of this equation must be a function of x only. The function µ ( x ) can be found by integrating the last equation.

Similarly, if , then we obtain an ordinary differential equation that determines the integrating factor µ :

where the right-hand side depends only on y. The function µ ( y ) is found by integrating this equation.

The new function z ( x,y ) can be, for example, of the type:

and so on.

What is important here is that the integrating factor µ ( x,y ) will be some function of one variable z :

and can be found from the differential equation:

It is assumed that the right-hand side of the equation depends only on z and that the denominator is nonzero.

Below we will consider some special cases of the equation.

for which an integrating factor can be found. General conditions for the existence of an integrating factor are derived in Lie group theory.

Solve the equation   (1 + y ² ) dx + xydy = 0 .

Let us first check that this equation is not an equation in total differentials:

      

As can be seen, the partial derivatives are not equal to each other, so the equation does not belong to the type of equations in total differentials. Let’s try to select an integrating factor to transform the equation to this type. Let’s calculate the function.

      

It can be seen that the expression.

      

depends only on the variable x. Therefore, the integrating factor will also depend only on x: µ = µ ( x ). We can find it from the equation:

      

Separating the variables and integrating, we obtain:

      

Let’s choose µ = x. Multiplying the original differential equation by µ = x, we obtain an equation in total differentials:

      

Indeed, we now have

      

Let’s solve the last equation. The function u ( x,y ) can be found from the system of equations:

      

From the first equation, it follows that.

      

We substitute this into the second equation to determine φ ( y ) :

      

It follows that φ ( y ) = C, where C is an arbitrary constant.

Thus, the general solution of the differential equation is determined by the implicit expression.

      

Solve the differential equation   ( x − cos y ) dx − sin y dy = 0 .

Applying our test for membership in total differential equations, we find:

      

Therefore, the given equation is not an equation in total differentials. Let us try to “construct” an integrating factor. Note that

      

and expression

      

will be a constant.

Therefore, we can look for the integrating factor as a function µ ( x ) by solving the corresponding equation:

      

Let us choose the function µ = e ^−x and verify that the original equation becomes an equation in total differentials after multiplication by µ = e ^−x :

      

Its general solution can be found from the system of equations:

      

In this case, it is more convenient to first integrate the second equation with respect to the variable y :

      

Substituting this into the first equation, we get

      

Integration by parts leads to the following result:

      

Thus, the general solution of the equation is described by the relation.

      

where C is an arbitrary real number.

Solve the differential equation   ( xy ² − 2 y ³ ) dx + (3 − 2 xy ² ) dy = 0 .

This equation is not an equation in total differentials, since

      

Let’s try to determine its general solution using an integrating factor. Let’s calculate the difference.

      

Note that the expression.

      

depends only on y. Therefore, the integrating factor µ will also be a function of one variable, y. We can find it from the equation.

      

Integrating, we find:

      

By choosing as the integrating factor and then multiplying the original differential equation by it, we obtain an equation in total differentials:

      

Indeed, it is now clear that.

      

Note that when multiplying by the integrating factor, we lost the solution y = 0. This can be proven by directly substituting the solution y = 0 into the original differential equation.

Now we find the function u from the system of equations:

      

From the first equation, it follows that.

      

From the second equation we find:

      

Thus, the given differential equation has the following solutions:

      

where C is an arbitrary constant.

Solve the equation   ( xy + 1) dx + x ² dy = 0 .

First, let’s make sure that the given equation is not an equation in total differentials:

      

The partial derivatives are not equal to each other. Therefore, the original equation is not a total differential equation. Let’s calculate the difference between the derivatives:

      

Let’s try applying an integrating factor in the form z = xy. Here we have:

      

Then

      

and, therefore, we obtain:

      

We see that the integrating factor depends only on the intermediate variable z :

      

This integrating factor can be determined from the last equation:

      

By choosing the function µ = e ^xy, we can transform the given differential equation into an equation in total differentials:

      

Let’s check this using the necessary and sufficient condition again:

      

So, it’s clear that the equation has now become a total differential equation. Its general solution is found from the system of equations:

      

We integrate the second equation with respect to the variable y (in this case, the variable x is considered constant):

      

Substituting into the first equation of the system, we obtain:

      

Therefore, the general solution of the differential equation is written in the form:

      

where C is an arbitrary real number.

Solve the equation y dx + ( x ² + y ² − x ) dy = 0 using the integrating factor   ( x,y ) = x ² + y ².

It is easy to see that the equation is not initially an equation in total differentials :

      

The difference of partial derivatives is equal to

      

Using the integrating factor   µ = z = x ² + y ² , we find:

      

Let’s calculate the following expression:

      

As a result, we obtain a differential equation for the function µ ( z ) :

      

We integrate and determine the function µ ( z ) :

      

You can choose an integrating factor . After multiplying the original differential equation by, it becomes a total differential equation:

      

The general solution u ( x,y ) = C is found from the following system of equations:

      

We integrate the first equation with respect to the variable x :

      

Substituting into the second equation, we obtain:

      

Thus, the general solution of the differential equation in implicit form is determined by the formula:

      

where C is an arbitrary constant.