Profile
Settings
Refer your friends
Sign out
In mathematics, the Ordinary differential equation (ODE) is an equation that consists of only one independent variable and one or more of its derivatives concerning the variable. Therefore, an ordinary differential equation will consist of one independent variable x, the real dependent variable y, with some of its derivatives. In this equation, the unknown is essentially a function and both the function and its derivatives appear in the equation. These are usually mathematical descriptions but they are also very much relevant in the core of a wide range of physical theories. Such as the classic equations for classic mechanics by Newton and Lagrange, Einstein’s theory of gravitation equation, Maxwell’s equation for classical electromagnetism, and Schrodinger’s equation for quantum mechanics. Other than these, this mathematical method has multiple applications in other subject areas as well.
Linear Ordinary Differential Equations
The linear ordinary differential equation can be derived in the form of dy/dx + Py = Q. in this equation, P and Q represent the numerical constants or the functions within x. The equation also consists of a derivative of y and a y. This differential is called the first-order linear differential equation as the differential is the first-order differentiation.
This linear differential equation is in y. by the same method, the linear equation in x can be written too. It will be dx/dy + Pıx = Qı
Formula- There are two formulas that are applied to linear differential equations to get solutions:
The solution of the differential equation dy/x +Py = Q is as follows. y.(I.F)=∫(Q.(I.F).dx)+Cy.(I.F)=∫(Q.(I.F).dx)+C. Here we have Integrating Factor (I.F) = e∫P.dx.
the general solution of the differential equation dx/y +Px = Q is as follows. x.(I.F)=∫(Q.(I.F).dy)+Cx.(I.F)=∫(Q.(I.F).dy)+C. Here we have Integrating Factor (I.F) = e∫P.dy
Steps to Solve Linear Differential Equation
In three simple steps, the linear differential equation can be solved:
Step 1: First, simplify the given differential equation and write it down in the form dy/dx + Py = Q, (in this equation, P and Q are numeric constants or functions in x).
Step 2: Figure out the Integrating Factor of the linear differential equation (IF) = e∫P.dxe∫P.dx.
Step 3: Finally, the solution of the linear differential equation can be written as; y(I.F)=∫(Q×I.F).dx+C.
Partial Differential Equations
In a partial differential equation, there is typically one or more than one partial derivative. It is a mathematical equation that has more than two independent variables, an unknown function that is dependent on the variables, along with partial derivatives of the unknown function concerning the independent variables. A solution or any form of a particular solution to a partial differential equation is a function that is responsible for solving the whole equation. The order of a partial differential equation is the order of the highest derivative involved in the same equation.
Conclusion
The ODE is an equation that consists of only one independent variable and one or more of its derivatives concerning the variable and an ODE will consist of one independent variable x, the real dependent variable y, with some of its derivatives. The ODEs can be classified into three types namely linear, non-linear and autonomous ODE. These ordinary differential equations are mathematical descriptions but they are also very much relevant in the core of a wide range of physical theories. Such as the classic equations for classic mechanics by Newton and Lagrange, Einstein’s theory of gravitation equation, Maxwell’s equation for classical electromagnetism, and
