Linear Ordinary Differential Equation

A linear ordinary differential equation (ODE) is an equation that involves only linear terms in the dependent variable and its differential coefficients. It can be written in the form:

a_n(x)y^n(x) + a_{n-1}(x)y^{n-1}(x) + ... + a_1(x)y'(x) + a_0(x)y(x) = g(x)

where y(x) is the dependent variable, y'(x) is its first derivative, y''(x) is its second derivative, and so on. The coefficients a_n(x), a_{n-1}(x), ..., a_1(x), a_0(x) and the function g(x) can all depend on the independent variable x.

Multiplication of Dependent Variable and its Differential Coefficients

In a linear ODE, the dependent variable and its differential coefficients are not multiplied together. This means that the terms in the equation involve either the dependent variable or its differential coefficients individually, but not their products.

For example, consider the following linear ODE:

a_2(x)y''(x) + a_1(x)y'(x) + a_0(x)y(x) = g(x)

In this equation, the dependent variable y(x) appears individually, the first derivative y'(x) appears individually, and the second derivative y''(x) appears individually. However, there are no terms where the dependent variable is multiplied by its differential coefficients, such as y(x)y'(x) or y(x)y''(x).

Degree of the Differential Coefficients

The degree of a differential coefficient in a linear ODE refers to the highest power to which the derivative appears. In other words, it represents the order of the differential coefficient.

In the given question, the correct answer is option 'A', which states that the dependent variable and its differential coefficients occur only in the first degree in a linear ODE. This means that the highest power to which the derivative appears is 1.

Conclusion

In a linear ODE, the dependent variable and its differential coefficients are not multiplied together and occur only in the first degree. This property distinguishes linear ODEs from nonlinear ODEs, where the dependent variable and its differential coefficients can be multiplied together and occur in higher degrees.