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 1a) \(y' = 3x\),
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 \(C = C_2 - C_1\).
References
Differential EquationsDifferential Equations - Runge Kutta Method