Solving Differential Equations

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself to its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, and other disciplines.

Differential equations take a form similar to:

$f(x) + f'(x) =0$

where $f'$ is "f-prime," the derivative of $f$. As you can see, such an equation relates a function $f(x)$ to its derivative. Solving the differential equation means solving for the function $f(x)$.

The "order" of a differential equation depends on the derivative of the highest order in the equation. The "degree" of a differential equation, similarly, is determined by the highest exponent on any variables involved. For example, the differential equation shown in is of second-order, third-degree, and the one above is of first-order, first-degree.

Example

Take the following differential equation:

$\displaystyle{\frac{d^2y}{dx^2} + xy = x^3y^3}$

Solving this equation shows that $f(x)$ is equal to the negative of its derivative; therefore, the function $f(x)$ must be $e^{-x}$, as the derivative of this function equals the negative of the original function. The derivative of $e^{-x}$ equals $e^{-x}$, so this must be the answer.

A complete solution contains the same number of arbitrary constants as the order of the original equation. (This is because, in order to solve a differential equation of the $n$th order, you will integrate $n$ times, each time adding a new arbitrary constant.) Since our example above is a first-order equation, it will have just one arbitrary constant in the complete solution. Therefore, the general solution is $f(x) = Ce^{-x}$, where $C$ stands for an arbitrary constant. You can see that the differential equation still holds true with this constant. For a specific solution, replace the constants in the general solution with actual numeric values.