Ordinary differential equation - Existence and nature of solutions

The problem of solving a differential equation is to find the function y whose derivatives satisfy the equation. For example, the differential equation has the general solution , where A, B are constants determined from boundary conditions. In general, an n-th order equation allows both x and y to be fixed, as well as all the nSee also:

Ordinary differential equation, Ordinary differential equation - Definition, Ordinary differential equation - General application, Ordinary differential equation - Existence and nature of solutions, Ordinary differential equation - Types of differential equations with some history, Ordinary differential equation - Homogeneous linear ODEs with constant coefficients, Ordinary differential equation - Linear ODEs with constant coefficients, Ordinary differential equation - Linear ODEs with variable coefficient, Ordinary differential equation - General solution method for first-order linear ODEs, Ordinary differential equation - Linear PDEs, Ordinary differential equation - First-order PDEs, Ordinary differential equation - Singular solutions, Ordinary differential equation - Reduction to quadratures, Ordinary differential equation - The Fuchsian theory, Ordinary differential equation - Lie's theory, Ordinary differential equation - Bibliography

Read more here: » Ordinary differential equation: Encyclopedia II - Ordinary differential equation - Existence and nature of solutions

In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is an equation that involves the derivatives of an unknown function of one variable. A simple example of an ordinary differential equation is , where is an unknown function, and is its derivative. See differential calculus and integral calculus for basic calculus background. Important theorems in the field of ODEs include broad existence and uniqueness theorems and for ODEs in the plane, the Poincaré-Ben ...

Including:

• Ordinary differential equation - Definition
• Ordinary differential equation - General application
• Ordinary differential equation - Existence and nature of solutions
• Ordinary differential equation - Types of differential equations with some history
• Ordinary differential equation - Linear ODEs with constant coefficients
• Ordinary differential equation - Linear ODEs with variable coefficient
• Ordinary differential equation - General solution method for first-order linear ODEs
• Ordinary differential equation - Linear PDEs
• Ordinary differential equation - First-order PDEs
• Ordinary differential equation - Singular solutions
• Ordinary differential equation - Reduction to quadratures
• Ordinary differential equation - The Fuchsian theory
• Ordinary differential equation - Lie's theory
• Ordinary differential equation - Bibliography

Read more here: » Ordinary differential equation: Encyclopedia - Ordinary differential equation

The influence of geometry, physics, and astronomy, starting with Newton and Leibniz, and further manifested through the Bernoullis, Riccati, and Clairaut, but chiefly through d'Alembert and Euler, has been very marked, and especially on the theory of linear partial differential equations with constant coefficients. Ordinary differential equation - Homogeneous linear ODEs with constant coefficients. The first method of integrating linear ordinary differential equations with constant coefficients is due to E ...

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Ordinary differential equation, Ordinary differential equation - Definition, Ordinary differential equation - General application, Ordinary differential equation - Existence and nature of solutions, Ordinary differential equation - Types of differential equations with some history, Ordinary differential equation - Homogeneous linear ODEs with constant coefficients, Ordinary differential equation - Linear ODEs with constant coefficients, Ordinary differential equation - Linear ODEs with variable coefficient, Ordinary differential equation - General solution method for first-order linear ODEs, Ordinary differential equation - Linear PDEs, Ordinary differential equation - First-order PDEs, Ordinary differential equation - Singular solutions, Ordinary differential equation - Reduction to quadratures, Ordinary differential equation - The Fuchsian theory, Ordinary differential equation - Lie's theory, Ordinary differential equation - Bibliography

Read more here: » Ordinary differential equation: Encyclopedia II - Ordinary differential equation - Types of differential equations with some history

An important special case is when the equations do not involve x. These differential equations may be represented as vector fields. This type of differential equation has the property that space can be divided into equivalence classes based on whether two points lie on the same solution curve. Since the laws of physics are believed not to change with time, the physical world is governed by such different ...

See also:

Ordinary differential equation, Ordinary differential equation - Definition, Ordinary differential equation - General application, Ordinary differential equation - Existence and nature of solutions, Ordinary differential equation - Types of differential equations with some history, Ordinary differential equation - Homogeneous linear ODEs with constant coefficients, Ordinary differential equation - Linear ODEs with constant coefficients, Ordinary differential equation - Linear ODEs with variable coefficient, Ordinary differential equation - General solution method for first-order linear ODEs, Ordinary differential equation - Linear PDEs, Ordinary differential equation - First-order PDEs, Ordinary differential equation - Singular solutions, Ordinary differential equation - Reduction to quadratures, Ordinary differential equation - The Fuchsian theory, Ordinary differential equation - Lie's theory, Ordinary differential equation - Bibliography

Read more here: » Ordinary differential equation: Encyclopedia II - Ordinary differential equation - General application

Let y represent an unknown function of x, and let denote the derivatives An ordinary differential equation (ODE) is an equation involving The order of a differential equation is the order n of the highest derivative that appears. If the highest derivative appears only in integer powers, then the degree of ...

See also:

Ordinary differential equation, Ordinary differential equation - Definition, Ordinary differential equation - General application, Ordinary differential equation - Existence and nature of solutions, Ordinary differential equation - Types of differential equations with some history, Ordinary differential equation - Homogeneous linear ODEs with constant coefficients, Ordinary differential equation - Linear ODEs with constant coefficients, Ordinary differential equation - Linear ODEs with variable coefficient, Ordinary differential equation - General solution method for first-order linear ODEs, Ordinary differential equation - Linear PDEs, Ordinary differential equation - First-order PDEs, Ordinary differential equation - Singular solutions, Ordinary differential equation - Reduction to quadratures, Ordinary differential equation - The Fuchsian theory, Ordinary differential equation - Lie's theory, Ordinary differential equation - Bibliography

Read more here: » Ordinary differential equation: Encyclopedia II - Ordinary differential equation - Definition