Table of Contents
Exact Differential Equation – Definition
Exact Differential Equation – Definition: A differential equation is a mathematical equation that relates some function with its derivatives. In mathematics, an exact differential equation is a differential equation in which the unknown function and its derivatives appear differently on each side of the equation, but are equal throughout the equation.
This means that the equation is an equality between a differential form of one degree higher than the unknown function, and a differential form of the same degree. The unknown function is a function of one or more independent variables, often represented by the letters x, y, z, and t.
Theorem
A differential equation is a mathematical equation that relates a function with one or more of its derivatives. In particular, an exact differential equation is a differential equation in which the unknown function and its derivatives are related by an equation that is exact.
Theorem: If a differential equation is exact, then it can be solved using the method of integration.
This theorem is important because it provides a way to solve differential equations that would otherwise be very difficult or impossible to solve. For example, consider the differential equation
y’ = x^2 + y^2
This differential equation is not exact, because the left-hand side is not equal to the right-hand side derivative. However, if we take the derivative of both sides, we get
y” = 2x + 2y y’
which is now an exact differential equation. Therefore, we can use the theorem to solve it.
First, we need to find the integrating factor, which is a function that when multiplied by the differential equation makes it exact. In this case, the integrating factor is
I = e^{\int 2x\,dx} = e^{x^2}
Now we can multiply both sides of the differential equation by the integrating factor to get
e^{x^2}y’ = e^{x^2}\left(x^2 + y^2\right)
Which is now an exact differential equation. Therefore, we can solve it using integration.
\begin{align*} \int e^{x^2}y’\,dx &= \int e^{x^2}\left(x^2 + y^2\right)\,dx \\ y &= \int e^{x^2}\,dx – \int x^2e^{x^2}\,dx + C \\ &= \frac{1}{2}e^{x^2} – \frac{1}{2}\int e^{x^2}\,dx + C \\ &= \frac{1}{2}e^{x^2} – \frac{1}{2}y + C \end{align*}
where C is an arbitrary constant. This is the general solution to the differential equation.
