Ordinary Differential Equations |
Equation: Equations describe the relations between the dependent and independent variables. An equal sign "=" is required in every equation. Differential Equation: Equations that involve dependent variables and their derivatives with respect to the independent variables are called differential equations. Ordinary Differential Equation: Differential equations that involve only ONE independent variable are called ordinary differential equations. Partial Differential Equation: Differential equations that involve two or more independent variables are called partial differential equations. |
Order and Degree |
Order: The order of a differential equation is the highest derivative that appears in the differential equation. Degree: The degree of a differential equation is the power of the highest derivative term. |
Linear, Non-linear, and Quasi-linear |
Linear: A differential equation is called linear if there are no multiplications among dependent variables and their derivatives. In other words, all coefficients are functions of independent variables. Non-linear: Differential equations that do not satisfy the definition of linear are non-linear. Quasi-linear: For a non-linear differential equation, if there are no multiplications among all dependent variables and their derivatives in the highest derivative term, the differential equation is considered to be quasi-linear. |
Homogeneous |
Homogeneous: A differential equation is homogeneous if every single term contains the dependent variables or their derivatives. Non-homogeneous: Differential equations which do not satisfy the definition of homogeneous are considered to be non-homogeneous. |
Solutions |
General Solution: Solutions obtained from integrating the differential equations are called general solutions. The general solution of a order ordinary differential equation contains arbitrary constants resulting from integrating times. Particular Solution: Particular solutions are the solutions obtained by assigning specific values to the arbitrary constants in the general solutions. Singular Solutions: Solutions that can not be expressed by the general solutions are called singular solutions. |
Conditions |
Initial Condition: Constrains that are specified at the initial point, generally time point, are called initial conditions. Problems with specified initial conditions are called initial value problems. Boundary Condition: Constrains that are specified at the boundary points, generally space points, are called boundary conditions. Problems with specified boundary conditions are called boundary value problems. |