Differential equations

Differential equations are mathematical equations that involve one or more derivatives of an unknown function. They are used to describe various phenomena in physics, engineering, and other scientific fields.

There are two main types of differential equations: ordinary differential equations (ODEs) and partial differential equations (PDEs). The main difference between them is the number of independent variables involved.

Ordinary Differential Equations (ODEs)

ODEs are differential equations that involve only one independent variable. In other words, the unknown function and its derivatives depend on a single variable. The general form of an ODE is:

F(x, y, y', y'', ..., y^(n)) = 0

where x is the independent variable, y is the dependent variable, y' represents the first derivative of y with respect to x, y'' represents the second derivative, and so on, up to the nth derivative.

Example of an ODE

One example of an ODE is the simple harmonic oscillator equation:

m * y'' + k * y = 0

where m is the mass of the object, k is the spring constant, and y represents the displacement of the object from its equilibrium position. This equation involves only one independent variable (time) and one dependent variable (displacement).

Why option 'C' is the correct answer?

In the given question, the option 'C' states that an ordinary differential equation has only one independent variable. This statement is true because an ODE involves a single independent variable and describes the relationship between the dependent variable and its derivatives with respect to that variable.

The other options - 'A', 'B', and 'D' - are incorrect because they either refer to the number of dependent variables or the number of independent variables, which is not the distinguishing factor for classifying an equation as ordinary or partial.

Conclusion

In summary, an ordinary differential equation is characterized by having only one independent variable. This classification is important in understanding the nature and behavior of the equation and allows for the use of specific techniques and methods to solve it.