Some of its examples are y'' + xy' - y sinx = x, y'' - 9x 2y' + 2e xy = 0, etc. The differential equation y'' + p(x)y' + q(x)y = f(x) is called a second order differential equation with variable coefficients if the functions p(x) and q(x) are not constant functions and are functions of x. Second Order Differential Equation with Variable Coefficients Some of its examples are y'' + y' - 6y = x, y'' - 9y' + 20y = sin x, etc. The differential equation y'' + p(x)y' + q(x)y = f(x) is called a second order differential equation with constant coefficients if the functions p(x) and q(x) are constants. Second Order Differential Equation with Constant Coefficients Non-homogeneous Second Order Differential EquationĪ differential equation of the form y'' + p(x)y' + q(x)y = f(x) is said to be a non-homogeneous second order differential equation if f(x) is not a zero function. Some of its examples are y'' + y' - 6y = 0, y'' - 9y' + 20y = 0, etc. Homogeneous Second Order Differential EquationĪ second order differential equation y'' + p(x)y' + q(x)y = f(x) is said to be a second order homogeneous differential equation if f(x) is a zero function and hence mathematically it of the form, y'' + p(x)y' + q(x)y = 0. Some of its examples are y'' + 6x = 5, y'' + xy' + y = 0, etc. Let us go through some special types of second order differential equations given below: Linear Second Order Differential EquationĪ linear second order differential equation is written as y'' + p(x)y' + q(x)y = f(x), where the power of the second derivative y'' is equal to one which makes the equation linear. These differential equations can be solved using the auxiliary equation. It can be of different types depending upon the power of the derivative and the functions involved. Solving Second Order Differential EquationįAQs on Second Order Differential EquationĪ second order differential equation is defined as a differential equation that includes a function and its second-order derivative and no other higher-order derivative of the function can appear in the equation. Second Order Differential Equation Definition What is a Second Order Differential Equation? We will also learn different methods to solve second order differential equations with constant coefficients using the various methods with the help of solved examples and finding the auxiliary equation. In this article, we will understand such differential equations in detail and their different types. The differential equation y'' + p(x)y' + q(x)y = 0 is called a second order differential equation with constant coefficients if the functions p(x) and q(x) are constants and it is called a second-order differential equation with variable coefficients if p(x) and q(x) are not constants. We can solve this differential equation using the auxiliary equation and different methods such as the method of undetermined coefficients and variation of parameters. Generally, we write a second order differential equation as y'' + p(x)y' + q(x)y = f(x), where p(x), q(x), and f(x) are functions of x. which indicates the second order derivative of the function. It includes terms like y'', d 2y/dx 2, y''(x), etc. Second order differential equation is a specific type of differential equation that consists of a derivative of a function of order 2 and no other higher-order derivative of the function appears in the equation.
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