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A linear differential equation is an equation that involves a function and its derivatives. It is called linear because the function and its derivatives are raised to the first power and there are no products of the function or its derivatives. It can be written in the form of y' + p(x)y = g(x), where y' represents the derivative of y, p(x) and g(x) are functions of x.
The main difference between a linear and a non-linear differential equation is that a linear differential equation involves a function and its derivatives raised to the first power, while a non-linear differential equation involves products, powers, or any other non-linear combinations of the function and its derivatives. This makes solving linear differential equations easier compared to non-linear ones.
The order of a linear differential equation is determined by the highest derivative that appears in the equation. For example, y'' + 3y' + 2y = 0 is a second-order linear differential equation because the highest derivative is y''. The order of a differential equation can also help determine the number of initial conditions needed to solve the equation.
The general method for solving a linear differential equation involves finding an integrating factor, multiplying both sides of the equation by it, and then integrating both sides. This will result in the solution of the differential equation. However, the specific method for solving a linear differential equation depends on its order and the type of function involved.
Linear differential equations are used in various fields of science and engineering to model physical systems and phenomena. Some common applications include population growth, radioactive decay, circuit analysis, and heat transfer. They are also used in economics and finance to model interest rates and stock prices.
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