What is ordinary differential equation? State any three differential equations.
Answers
Answer :-
Ordinary differential equations. An ordinary differential equation (ODE) is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. The unknown function is generally represented by a variable (often denoted y), which, therefore, depends on x.
This is an equation which relates to maths-physics both.
A Differential equation is a mathematics related equation which relates some function with it derivatives. In case of applications the function usually represents their rates of change, and the equation defines the relationship between the two. Because such relations are extremely common and differential equations play a very important role.
In pure mathamatics, differential equations are studied from different perspectives.
In some cases, this differential equation (called an equation of motion) may be solved explicitly.
The study of these differential equations is a wide field in pure and applied mathematics, physics, and engineering. All the above disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, whereaso applied mathematics emphasizes the rigorous justification of the methods for approximating solutions.
The three examples of differential equation are:-
☆ A Separable first-order ordinary differential equation- A separable first-order differential equation is said to be separable if, after solving it for the derivative, dy dx = F(x, y) , the right-hand side can then be factored as “a formula of just x ” times “a formula of just y ”, F(x, y) = f (x)g(y) .
☆ A Separable (homogeneous) first-order linear ordinary differential equation- A first‐order differential equation is said to be homogeneous if M( x,y) and N( x,y) are both homogeneous functions of the same degree. ... The substitution y = xu (and therefore dy = xdu + udx) transforms a homogeneous equation into a separable one. Example 7: Solve the equation ( x 2 – y 2) dx + xy dy = 0.
☆A Second-order linear ordinary differential equation- So for an ordinary differential equation in which is a constant, the solution is given by solving the second-order linear ODE with constant coefficients. (23) for , where is defined as above. A linear second-order homogeneous differential equation of the general form.
___________________________________