Differential equations
Differential equations are of many types and are classified according to their categories. The classifications of differential equations as linear or non-linear, autonomous or nonautonomous, homogenous or non-homogenous, ordinary or partial, scalar or vector, and the order of the differential equation in question is shown below.
Equation 1
The above equation is a non-linear differential equation because the equation does not have any arbitrary dependence on an independent variable. There is only one variable in the overall equation. The equation is non-homogeneous since it cannot be put in the form:
The equation is also autonomous because it does not change with time. The equation, however, is an ordinary differential equation and it falls in the category of second differential equations since there is a second power on the function.
Equation 2
The equation is linear since there is evidence of dependence between one variable and the other. The equation is nonautonomous because it is likely to change with time. It is homogeneous, and similarly, it is a partial differential equation. It is a scalar, and it falls in the category of second-order differential equations since the highest power is two.
Equation 3
The equation is linear, and it is also not autonomous. It is a non-homogeneous equation, but it is an ordinary differential equation. Finally, the equation is scalar, and it is a first-order equation.
Equation 3
The equation is non-linear since it does not have an independent variable. It is also an autonomous equation since it is not likely to change over time. It is an ordinary differential equation, a vector, and a third-order differential equation.
Equation 4
The equation is non-linear, it is autonomous since there is no dependence between the dependent and independent variable, and therefore, it is also non-homogeneous. It is an ordinary differential equation, a scalar, and it is a first-order equation.
Equation 5
The equation is linear since there is evidence of the inter-dependence between the dependent and independent variables. It is autonomous because it is not likely to vary with time. It is also homogeneous, an ordinary differential equation, a scalar, and a seventh order differential equation.
Equation 7
The equation is non-linear, autonomous because there is no dependence on variables. The equation is also non-homogeneous, it is a partial ordinary differential equation, a scalar, and is a tenth order differential equation.
Equation 8
The equation is linear since the dependent variable is affected by the independent variable. The equation is not autonomous, and at the same time, it is homogenous since it can be written in the form . The equation is an ordinary differential equation, it is a scalar, and finally, it is a second-order differential equation.
Equation 9
The equation is non-linear since there is no evidence of interdependence between dependent and independent variables. It is autonomous since it does not change with the variation in time. The equation is not homogenous, it is a scalar, and lastly, it is a first-order differential equation.
Equation 10
The equation is non-linear, and therefore, it is autonomous because there is no dependence between the dependent and independent variables. It is not homogenous and a vector because of the vector function that is in an Euclidean space. The equation falls in the category of first-order differential equations because its highest power is one.