A second order linear differential equation is a mathematical expression that describes the relationship between a function and its derivatives. It is in the form of y'' + p(x)y' + q(x)y = g(x), where y is the unknown function, x is the independent variable, and p(x), q(x), and g(x) are known functions.
The main difference between first and second order differential equations is the highest derivative that appears in the equation. A first order equation has the first derivative as the highest, while a second order equation has the second derivative as the highest. Additionally, second order equations are more complex and require two initial conditions to find a unique solution, while first order equations only require one initial condition.
The general solution to a second order linear differential equation is a family of functions that satisfy the equation. It is in the form of y = c1y1(x) + c2y2(x), where y1(x) and y2(x) are linearly independent solutions and c1 and c2 are arbitrary constants.
To solve a second order linear differential equation with constant coefficients, you can use the method of undetermined coefficients or the method of variation of parameters. The first method involves assuming a particular solution and then finding the general solution using the characteristic equation. The second method involves finding a particular solution by varying the constants in the general solution.
Second order linear differential equations have many practical applications in physics, engineering, and other fields. They can be used to model the motion of a spring, the growth of a population, the vibrations of a guitar string, and many other natural phenomena. They are also used in electrical and mechanical systems to describe the relationship between variables such as current, voltage, and position.