Ordinary Differential Equation
An ordinary differential equation (ODE) is a type of equation that involves ordinary derivatives, not partial derivatives. It typically includes variables and a derivative of the dependent variable with respect to the independent variable. Such equations contain at least one derivative of an unknown function, which can be either an ordinary derivative or a partial derivative.
Ordinary differential equations specifically involve ordinary derivatives, and they are commonly referred to simply as “differential equations.”
Definition of Ordinary Differential Equation
An ordinary differential equation is a mathematical equation that involves the derivatives of an unknown function with respect to a single independent variable. It describes the relationship between function and its derivatives, commonly used to model various dynamic systems in physics, engineering, and other scientific fields.
General Form of Ordinary Differential Equations
The general form of an ordinary differential equation (ODE) is represented as
F(x, y, y’, y”, …) = 0
Where (x) is the independent variable, (y) is the dependent variable, and (y’), (y”), etc., denote the first, second, and higher order derivatives of (y) with respect to (x) respectively.
Examples of Ordinary Differential Equation
Some examples of ODE are:
- (dy/dx = 2x): This equation represents the first-order ordinary differential equation where the derivative of ( y ) with respect to ( x ) is equal to ( 2x ).
- [Tex]\bold{\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 0}[/Tex]: This is a second-order ordinary differential equation where the second derivative of ( y ) with respect to ( x ), plus three times the first derivative of ( y ) with respect to (x), plus two times (y), equals zero.
- [Tex]\bold{\frac{d^2y}{dx^2} + y\frac{dy}{dx} = 0}[/Tex]: Another example of a second-order ordinary differential equation, where the second derivative of ( y ) with respect to ( x ), plus (y) times the first derivative of (y) with respect to (x), equals zero.
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Ordinary Differential Equations
Ordinary Differential Equations(ODE) is the mathematical equation that describe how a function’s rate of change relates to its current state. It involves a single independent variable and its derivatives.
Ordinary Differential Equations
Let’s know more about Ordinary Differential Equations, it’s types, order and degree of Ordinary differential equation in detail below.
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