What is a differential equation?

An equation that involves one or more derivatives of an unknown function is called a **differential equation**. The **order** of the highest derivative included in a differential equation defines the **order** of this equation.

a) \(y' = 3x\),

the order of the highest derivative is 1 (\(y'\)), so the order of this differential equation is 1.

b) \(y'' + y' + y = 3x\),

the order of the highest derivative is 2 (\(y''\)), so the order of this differential equation is 2.

c) \(-2y''' + y'' + y^4 = 3x\),

the order of the highest derivative is 3 (\(y'''\)), so the order of this differential equation is 3.

\(y = f(x)\) is a **solution** of a differential equation if the equation is satisfied upon substitution of \(y\) and its derivatives into the differential equation.

Example 2

Verify that \(y = Ce^{4x} + e^{3x}\), where \(C\) is a constant, is a solution to the differential equation

\[
y' - 4y = -e^{3x}
\]

\(y'\) is given by

\(y' = 4Ce^{4x} + 3e^{3x}\)

We now substitute \(y'\) and \(y\) into the left side of the equation and simplify

\(y' - 4y = 4Ce^{4x} + 3e^{3x} - 4(Ce^{4x} + e^{3x})\)

\(= 4Ce^{4x} + 3e^{3x} - 4Ce^{4x} - 4e^{3x}\)

\(= 4Ce^{4x} - 4Ce^{4x} + e^{3x}(3 - 4)\)

\(= -e^{3x}\)

Which is equal to the left side of the given equation and therefore \(y = Ce^{4x} + e^{3x}\) is a solution to the differential equation \(y' - 4y = -e^{3x}\).

Most of the work on differential equations consists in **solving** these equations. For example, to **solve** the following differential example

\[
y' = 2x
\]

Let us integrate both sides of the given equation as follows

\[
\int y' \, dx = \int 2x \, dx
\]

which gives

\(y + C_1 = x^2 + C_2\)

where \(C_1\) and \(C_2\) are constants of integration. The solution \(y\) of the above equation is given by: \(y = x^2 + C\), where \(C = C_2 - C_1\).

Differential Equations - Runge Kutta Method