General Terms of Ordinary Differential Equations
engineering fundamentals General Terms of Ordinary Differential Equations
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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 nth order ordinary differential equation contains n arbitrary constants resulting from integrating n 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.

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