Equations in total differentials

Differential equation of the form

is called an equation in total differentials if there exists a function of two variables u ( x,y ) with continuous partial derivatives such that the expression

The general solution of the equation in total differentials is determined by the formula.

where C is an arbitrary constant.

Let functions P ( x,y ) and Q ( x,y ) have continuous partial derivatives in some domain D . The differential equation P ( x,y ) dx + Q ( x,y ) dy = 0 will be an equation in total differentials if and only if the equality holds:

1. First, let us verify that the differential equation is a total differential equation using the necessary and sufficient condition :
   
2. Then we write down a system of two differential equations that define the function u ( x,y ) :
   
3. We integrate the first equation with respect to the variable x. Instead of the constant C, we write an unknown function depending on y :
   
4. Differentiating with respect to the variable y, we substitute the function u (x, y) into the second equation:
   

   From here we obtain an expression for the derivative of the unknown function φ ( y ) :

   
5. Integrating the last expression, we find the function φ ( y ) and, consequently, the function u ( x,y ) :
   
6. The general solution of the equation in total differentials is written as:
   

Note: In step 3, instead of integrating the first equation with respect to the variable x, we can integrate the second equation with respect to the variable y. After integration, we need to determine the unknown function ψ ( x ).

Solve the differential equation   2 xydx + ( x ² + 3 y ² ) dy = 0 .

This equation is an equation in total differentials, since the corresponding partial derivatives are equal to:

      

Let us write the following system of differential equations to determine the function u ( x,y ) :

      

Integrating the first equation with respect to x, we obtain:

      

We substitute the expression for u (x, y) into the second equation:

      

Integrating the last equation, we find the unknown function φ ( y ) :

      

so that the general solution of this equation in total differentials has the form:

      

where C is an arbitrary constant.

Find the solution of the differential equation   (6 x ² − y + 3) dx + (3 y ² − x − 2) dy = 0 .

Let’s check whether this equation is an equation in total differentials:

      

As you can see, we have an equation in total differentials. Let’s write a system of equations to determine the function u ( x,y ) :

      

We integrate the first equation with respect to the variable x, assuming that y is a constant. As a result, we obtain:

      

Here we have introduced a continuous differentiable function φ ( y ) instead of the constant C.
Let us substitute the function u (x, y) into the second equation:

      

We obtain an equation for the derivative φ’ ( y ) :

      

By integrating, we find the function φ ( y ) :

      

Thus, the function u (x, y) is defined by the formula.

      

Therefore, the general solution of the equation is described by the following implicit expression:

      

where C is an arbitrary real number.

Solve the differential equation e ^y dx + (2 y + xe ^y ) dy = 0.

First, let’s check that this equation is an equation in total differentials:

      

It is clear that . Let us further find the function u ( x,y ) from the system of equations:

      

Hence,

      

Now we differentiate this expression with respect to the variable y and equate it to . As a result, we obtain an expression for the derivative φ’ ( y ) :

      

Thus we find φ ( y ) and the entire function u ( x,y ) :

      

Therefore, the general solution of the differential equation is written as:

      

Solve the equation   (2 xy − sin x ) dx + ( x ² – cos y ) dy = 0 .

This equation is an equation in total differentials, since

      

Let’s find the function u (x, y) from the system of two equations:

      

Integrating the first equation with respect to the variable x, we obtain:

      

Substituting into the second equation, we have:

      

Hence,

      

Then the function u ( x,y ) is defined by the expression.

      

and the general solution of the differential equation is described by an implicit formula

      

Solve the equation

First, let’s find out whether we are dealing with an equation in total differentials:

      

As can be seen, . Therefore, this is an equation in total differentials. Let us find a function u ( x,y ) satisfying the system of equations:

      

We integrate the first equation:

      

where φ ( y ) is some unknown function depending on y. We will determine it later.

Substitute the result into the second equation of the system:

      

Integrating the last expression, we find the function φ ( y ) :

      

where C is a constant.

Thus, the general solution of this differential equation is described by the equation:

      

Solve the differential equation with the initial condition y (1) = 1.

Let us check that the equation is an equation in total differentials, having first transformed it into standard form:

      

The partial derivatives will be equal to

      

Therefore, we are dealing with an equation in total differentials. Therefore, we will now write the following system of equations to determine the function u ( x,y ) :

      

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

      

Now let’s differentiate this expression with respect to the variable x :

      

Thus, the general solution of the differential equation is implicitly determined by the expression:

      

Let us now find a particular solution that satisfies the initial condition y (1) = 1 . Substituting the initial values, we determine the constant C :

      

Therefore, a particular solution of this Cauchy problem has the form: