Exact differential equation

An exact differential equation is a first order differential equation that has the form:

$M(t,y)+N(t,y)y'=0\,$ or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M(t,y) dt + N(t,y) dy = 0\,}

and is also exact. This means that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial M}{\partial y} = \frac{\partial N}{\partial t}}

If a first order differential equation is exact, then a conservative field exists and a scalar potential can be defined.

Solution

To find the solution of this equation, we consider the fact that the equation has the form of an exact differential of a function φ(y,t). The total differential of φ(y,t) is then:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\phi (y,t) = \frac{\partial \phi}{\partial y} dy + \frac{\partial \phi}{\partial t} dt }

Comparing to our original differential equation, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \phi}{\partial t} = M} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \phi}{\partial y} = N} and also that this implies that dφ(y,t)=0 and φ(y,t)=constant. To solve the equation, we integrate both sides of the total differential with respect to one of the variables, say t:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(y,t) = \int M(t,y) \, dt + F(y) }

where F(y) is a "constant of integration". (In reality, F(y) is a function, but when we differentiate it with respect t, we treat y as constant and so its derivative becomes zero.) In order to solve for F(y), we differentiate φ(y,t) with respect to the other variable, in this case y, and set the result equal to N(t,y).

Non-Exact Case

If the two functions, M(t,y) and N(t,y), instead have the property:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial t} }

then the equation is said to be "inexact" and the above method cannot be used directly to solve it. However, it can be used if a function, μ(t,y) can be found such that, when the differential equation is multiplied by μ(t,y), it becomes exact. Such a function is called an "integrating function". This would transform the condition for exactness to:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial}{\partial y} \left( \mu(y,t) M \right) = \frac{\partial}{\partial t} \left( \mu(y,t) N \right) }

meaning that the integrating function can be found as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu = \frac{N\frac{\partial \mu}{\partial t}-M\frac{\partial \mu}{\partial y}}{\frac{\partial M}{\partial y}-\frac{\partial N}{\partial t}} }

This isn't particularly useful by itself as we still have to solve for μ as a function of two variables. Often, the integrating function can be expressed as only one of the two variables, meaning we would only need to solve an ordinary differential equation rather than a partial differential equation. Otherwise, the integrating factor can sometimes be guessed or another method of finding a solution can be employed.

Example

Consider the equation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t \frac{dy}{dt}+3t+y=0 }

Here we have M(t,y)=3t+y and N(t,y)=t. As:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial M}{\partial y} = 1 = \frac{\partial N}{\partial t} }

the equation is exact. Integrating M(t,y) with respect to t produces:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(t,y) = \int (3t+y) \, dt + F(y) = \frac{3t^2}{2} + ty + F(y) }

and so:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \phi}{\partial y} = N(t,y) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t + F'(y) = t }

Therefore F'(y)=0 which means f(y) is a constant, a. Therefore the solution is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(t,y) = \frac{3t^2}{2} + ty = \text{constant} }

where the constant a has been combined with the constant of the condition φ(t,y)=constant.

See also

• Differential equations
• Separation of variables