Introduction to Differential Equations

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.

Examples and Detailed Solutions

Example 1

\(y = f(x)\) 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 \(C = C_2 - C_1\).

References

Differential Equations
Differential Equations - Runge Kutta Method