General and Particular Solutions of a Differential Equation

A differential equation involves derivatives of an unknown function, and its solution is usually a function rather than a single number.

This topic explains the meaning of a solution of a differential equation and distinguishes between a general solution and a particular solution.

For a differential equation, a solution is a function that makes the left-hand side equal to the right-hand side when the function and its required derivatives are substituted. If y = ϕ(x) satisfies the differential equation, then the curve represented by y = ϕ(x) is called the solution curve or integral curve.

A solution containing arbitrary constants is called the general solution of a differential equation.

A solution obtained from the general solution by assigning specific values to the arbitrary constants is called a particular solution.

Verification of a particular function

Verify that \[y = e^{-3x}\] is a solution of \[\frac{d^2y}{dx^2} + \frac{dy}{dx} - 6y = 0\].

Step 1: Differentiate once

\[\frac{dy}{dx} = -3e^{-3x}\]

Step 2: Differentiate again

\[\frac{d^2y}{dx^2} = 9e^{-3x}\]

Step 3: Substitute into the differential equation

\[\frac{d^2y}{dx^2} + \frac{dy}{dx} - 6y = 9e^{-3x} - 3e^{-3x} - 6e^{-3x} = 0\]

\[9e^{-3x} - 9e^{-3x}\] = 0

Conclusion: Hence, \[y = e^{-3x}\] is a solution of the given differential equation.

• A differential equation contains derivatives of an unknown function.

• Its solution is generally a function, not a single number.

• The graph of the solution function is called the solution curve or integral curve.

• A general solution contains arbitrary constants.

• A particular solution is obtained by assigning fixed values to those constants.

• To verify a solution, substitute the function and its derivatives into the equation and check whether LHS = RHS.