A differential equation is an equation that involves an unknown function and one or more of its derivatives. If \( y = f(x) \), then expressions such as \( y' \), \( y'' \), or higher derivatives may appear in the equation.
Order of a Differential Equation
The order of a differential equation is defined as the order of the highest derivative appearing in the equation.
Examples: Determining the Order
-
\( y' = 3x \)
The highest derivative is \( y' \). Therefore, this is a first-order differential equation. -
\( y'' + y' + y = 3x \)
The highest derivative is \( y'' \). Therefore, this is a second-order differential equation. -
\( -2y''' + y'' + y^4 = 3x \)
The highest derivative is \( y''' \). Therefore, this is a third-order differential equation.
Verifying a Solution
A function \( y = f(x) \) is called a solution of a differential equation if, after substituting \( y \) and its derivatives into the equation, both sides become equal.
Example: Verifying a Solution
Verify that \( y = Ce^{4x} + e^{3x} \) is a solution of the differential equation:
\[ y' - 4y = -e^{3x} \]First compute the derivative:
\[ y' = 4Ce^{4x} + 3e^{3x} \]Substitute \( y \) and \( y' \) into the left side:
\[ \begin{aligned} y' - 4y &= 4Ce^{4x} + 3e^{3x} - 4(Ce^{4x} + e^{3x}) \\ &= 4Ce^{4x} + 3e^{3x} - 4Ce^{4x} - 4e^{3x} \\ &= e^{3x}(3 - 4) \\ &= -e^{3x} \end{aligned} \]This matches the right side of the differential equation. Therefore, \( y = Ce^{4x} + e^{3x} \) is indeed a solution.
Solving a Simple Differential Equation
Most applications of differential equations involve finding the unknown function.
Example: Finding the General Solution
Solve:
\[ y' = 2x \]Integrate both sides:
\[ \int y' \, dx = \int 2x \, dx \]This gives:
\[ y + C_1 = x^2 + C_2 \]Combining constants:
\[ y = x^2 + C \]where \( C = C_2 - C_1 \) is an arbitrary constant.