# Exact Differential Equations

## Definition of Exact Equation

A differential equation of type

is called an exact differential equation if there exists a function of two variables \(u\left( {x,y} \right)\) with continuous partial derivatives such that

The general solution of an exact equation is given by

where \(C\) is an arbitrary constant.

## Test for Exactness

Let functions \(P\left( {x,y} \right)\) and \(Q\left( {x,y} \right)\) have continuous partial derivatives in a certain domain \(D.\) The differential equation \(P\left( {x,y} \right)dx + Q\left( {x,y} \right)dy = 0\) is an exact equation if and only if

## Algorithm for Solving an Exact Differential Equation

- First it's necessary to make sure that the differential equation is exact using the test for exactness:
\[\frac{{\partial Q}}{{\partial x}} = \frac{{\partial P}}{{\partial y}}.\]
- Then we write the system of two differential equations that define the function \(u\left( {x,y} \right):\)
\[\left\{ \begin{array}{l} \frac{{\partial u}}{{\partial x}} = P\left( {x,y} \right)\\ \frac{{\partial u}}{{\partial y}} = Q\left( {x,y} \right) \end{array} \right..\]
- Integrate the first equation over the variable \(x.\) Instead of the constant \(C,\) we write an unknown function of \(y:\)
\[u\left( {x,y} \right) = \int {P\left( {x,y} \right)dx} + \varphi \left( y \right).\]
- Differentiating with respect to \(y,\) we substitute the function \(u\left( {x,y} \right)\)into the second equation:
\[\frac{{\partial u}}{{\partial y}} = \frac{\partial }{{\partial y}}\left[ {\int {P\left( {x,y} \right)dx} + \varphi \left( y \right)} \right] = Q\left( {x,y} \right).\]From here we get expression for the derivative of the unknown function \({\varphi \left( y \right)}:\)\[\varphi'\left( y \right) = Q\left( {x,y} \right) - \frac{\partial }{{\partial y}}\left( {\int {P\left( {x,y} \right)dx} } \right).\]
- By integrating the last expression, we find the function \({\varphi \left( y \right)}\) and, hence, the function \(u\left( {x,y} \right):\)
\[u\left( {x,y} \right) = \int {P\left( {x,y} \right)dx} + \varphi \left( y \right).\]
- The general solution of the exact differential equation is given by
\[u\left( {x,y} \right) = C.\]

### Note:

In Step \(3,\) we can integrate the second equation over the variable \(y\) instead of integrating the first equation over \(x.\) After integration we need to find the unknown function \({\psi \left( x \right)}.\)

## Solved Problems

Click or tap a problem to see the solution.

### Example 1

Solve the differential equation \[2xydx + \left( {{x^2} + 3{y^2}} \right)dy = 0.\]

### Example 2

Find the solution of the differential equation

### Example 1.

Solve the differential equation \[2xydx + \left( {{x^2} + 3{y^2}} \right)dy = 0.\]

Solution.

The given equation is exact because the partial derivatives are the same:

We have the following system of differential equations to find the function \(u\left( {x,y} \right):\)

By integrating the first equation with respect to \(x,\) we obtain

Substituting this expression for \(u\left( {x,y} \right)\) into the second equation gives us:

By integrating the last equation, we find the unknown function \({\varphi \left( y \right)}:\)

so that the general solution of the exact differential equation is given by

where \(C\) is an arbitrary constant.

### Example 2.

Find the solution of the differential equation

Solution.

We check this equation for exactness:

Hence, the given differential equation is exact. Write the system of equations to determine the function \(u\left( {x,y} \right):\)

Integrate the first equation with respect to the variable \(x\) assuming that \(y\) is a constant. This produces:

Here we introduced a continuous differentiable function \(\varphi \left( y \right)\) instead of the constant \(C.\)

Plug in the function \(u\left( {x,y} \right)\) into the second equation:

We get equation for the derivative \(\varphi'\left( y \right):\)

Integrating gives the function \(\varphi \left( y \right):\)

So, the function \(u\left( {x,y} \right)\) is given by

Hence, the general solution of the equation is defined by the following implicit expression:

where \(C\) is an arbitrary real number.