The Solution of a Differential Equation with no Arbitrary Constants

A differential equation is an equation that relates a function to its derivatives. The solution of a differential equation is a function that satisfies the equation when substituted into it. In general, a differential equation has a set of solutions called the general solution, which contains arbitrary constants. However, there are cases where a differential equation has a solution that contains no arbitrary constants.

Particular Solution

A particular solution of a differential equation is a specific solution that satisfies the equation without containing any arbitrary constants. It is obtained by finding a function that satisfies the given differential equation without any additional conditions or restrictions. A particular solution represents one specific solution to the differential equation.

General Solution

The general solution of a differential equation contains arbitrary constants. It represents a family of solutions that satisfy the equation. These arbitrary constants can take on any value, allowing the general solution to encompass all possible solutions to the differential equation. The general solution represents an infinite set of solutions.

Primitive

In the context of differential equations, the term "primitive" is not commonly used. It is more commonly used in calculus to refer to the antiderivative of a function. The antiderivative is the reverse process of differentiation and represents a family of functions that differ by a constant.

Explanation of the Correct Answer

The correct answer to the given question is option 'A', particular solution. A particular solution of a differential equation contains no arbitrary constants and represents a specific solution that satisfies the equation. It is different from the general solution, which includes arbitrary constants and represents a family of solutions.

When solving a differential equation, it is important to consider whether the solution should be a particular solution or the general solution. The presence of arbitrary constants in the general solution allows for more flexibility and the inclusion of a wider range of possible solutions. However, in some cases, finding a particular solution that satisfies the equation without any arbitrary constants can be sufficient for the given problem or context.