The ordinary differential equation has infinitely many solutions. Solution of Ordinary Differential Equationsįor a given ordinary differential equation, y = φ(x) the solution curve (integral curve) is called the solution of the ordinary differential equation.

Some of the examples of linear differential equation in y are dy/dx + y = cos x, dy/dx + (-2y)/x = x 2.e -x and the examples of linear differential equation in x are dx/dy + x = sin y, dx/dy + x/y = ey. The differential is a first-order differentiation and is called the first-order linear differential equation. The linear differential equation in x is dx/dy + P 1x = Q 1. Similarly, we can write the linear differential equation in x also. This is referred to as a linear differential equation in y.

Here P and Q in this differential equation are either numeric constants or functions of x. The standard form of a linear differential equation is dy/dx + Py = Q, and it contains the variable y, and its derivatives.

Linear differential equation is an equation having a variable, a derivative of this variable, and a few other functions. One of the types of a non-homogeneous differential equation is the linear differential equation, which is similar to the linear equation. For example, xy(dy/dx) + y 2 + 2x = 0 is not a homogeneous differential equation. Note: xy(dy/dx) + y 2 + 2x = 0 is not a homogeneous differential equation Non-Homogeneous Differential EquationĪ differential equation in which the degree of all the terms is not the same is known as a nonhomogeneous differential equation. Some of the examples of homogeneous hifferential equations are as follows. In general they can be represented as P(x,y)dx + Q(x,y)dy = 0, where P(x,y) and Q(x,y) are homogeneous functions of the same degree. Homogeneous Differential EquationĪ differential equation in which the degree of all the terms is the same is known as a homogeneous differential equation. Let us check more about each of these two types of differential equations. The ordinary differential equations are broadly classified as homogeneous differential equations and non-homogeneous differential equations. Further, if a differential equation is not expressible in terms of a polynomial equation having the highest order derivative as the leading term, then that degree of the differential equation is not defined. The order and degree of a differential equation are always positive integers. Here the order of the differential equation is 4 and the degree is 3. To find the degree of the differential equation, we need to have a positive integer as the index of each derivative. The degree of the differential equation is the power of the highest ordered derivative present in the equation. If a differential equation is expressible in a polynomial form, then the integral power of the highest order derivative that appears is called the degree of the differential equation. Degree of Ordinary Differential Equations It is represented as d/dx(dy/dx) = d 2y/dx 2 = f”(x) = y”. Second-Order Differential Equation: The equation which includes second-order derivative is the second-order differential equation. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’ All the linear equations in the form of derivatives are in the first order. In these differential equations, the highest derivatives are of first, fourth and third order respectively and hence their orders are 1, 4, and 3 respectively.įirst Order Differential Equation: It is the first-order differential equation that has a degree equal to 1. Consider the following differential equations, dy/dx = e x, (d 4y/dx 4) + y = 0, (d 3y/dx 3) + x 2(d 2y/dx 2) = 0. The order of a differential equation is the order of the highest derivative of the dependent variable with respect to the independent variable. The two important aspects of ordinary differential equations, is the order and the degree of the differential equation. Order and Degree of Ordinary Differential Equation

Solution of ode