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## Classification of Differential Equations

We consider two methods of solving linear differential equations of first order:. This method is similar to the previous approach. The described algorithm is called the method of variation of a constant. Of course, both methods lead to the same solution. We will solve this problem by using the method of variation of a constant. ## Differential equation

In this chapter we will look at solving first order differential equations. The most general first order differential equation can be written as,. What we will do instead is look at several special cases and see how to solve those. We will also look at some of the theory behind first order differential equations as well as some applications of first order differential equations. Below is a list of the topics discussed in this chapter. Linear Equations — In this section we solve linear first order differential equations, i. We give an in depth overview of the process used to solve this type of differential equation as well as a derivation of the formula needed for the integrating factor used in the solution process.

In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering , physics , economics , and biology. Mainly the study of differential equations consists of the study of their solutions the set of functions that satisfy each equation , and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

While differential equations have three basic types — ordinary ODEs , partial PDEs , or differential-algebraic DAEs , they can be further described by attributes such as order, linearity, and degree. The solution method used by DSolve and the nature of the solutions depend heavily on the class of equation being solved. The order of a differential equation is the order of the highest derivative in the equation. A differential equation is linear if the equation is of the first degree in and its derivatives, and if the coefficients are functions of the independent variable. It should be noted that sometimes the solutions to fairly simple nonlinear equations are only available in implicit form. In these cases, DSolve returns an unevaluated Solve object. Linear Differential Equations. A first order differential equation y = f(x, y) is a linear equation if the function f is a “linear” expression in y. That is, the equation is​.

## First-order first-degree differential equation

Definition Example The general first order equation is rather too general, that is, we can't describe methods that will work on them all, or even a large portion of them. We can make progress with specific kinds of first order differential equations.

Methods of solution. Separation of variables. Homogeneous, exact and linear equations. ## Jennifer K.

EXACT DIFFERENTIAL EQUATION. Let M(x,y)dx + N(x,y)dy = 0 be a first order and first degree differential equation where M and N are real valued functions.