To choose one solution, more information is needed. Usually a given differential equation has an infinite number of solutions, so it is natural to ask which one we want to use. Passing through the point ( 1, 7 ), ( 1, 7 ), given that y = 2 x 2 + 3 x + C y = 2 x 2 + 3 x + C is a general solution to the differential equation. In fact, there is no restriction on the value of C C it can be an integer or not.) ( Note: in this graph we used even integer values for C C ranging between −4 −4 and 4. A graph of some of these solutions is given in Figure 4.2. This is an example of a general solution to a differential equation. It can be shown that any solution of this differential equation must be of the form y = x 2 + C. The reason is that the derivative of x 2 + C x 2 + C is 2 x, 2 x, regardless of the value of C. What if the last term is a different constant? Will this expression still be a solution to the differential equation? In fact, any function of the form y = x 2 + C, y = x 2 + C, where C C represents any constant, is a solution as well. The only difference between these two solutions is the last term, which is a constant. We already noted that the differential equation y ′ = 2 x y ′ = 2 x has at least two solutions: y = x 2 y = x 2 and y = x 2 + 4. ( x 4 − 3 x ) y ( 5 ) − ( 3 x 2 + 1 ) y ′ + 3 y = sin x cos x ( x 4 − 3 x ) y ( 5 ) − ( 3 x 2 + 1 ) y ′ + 3 y = sin x cos x General and Particular Solutions What function has a derivative that is equal to 3 x 2 ? 3 x 2 ? One such function is y = x 3, y = x 3, so this function is considered a solution to a differential equation. Therefore we can interpret this equation as follows: Start with some function y = f ( x ) y = f ( x ) and take its derivative. Furthermore, the left-hand side of the equation is the derivative of y. There is a relationship between the variables x x and y : y y : y is an unknown function of x. General Differential EquationsĬonsider the equation y ′ = 3 x 2, y ′ = 3 x 2, which is an example of a differential equation because it includes a derivative.
#Diff eq how to#
In this section we study what differential equations are, how to verify their solutions, some methods that are used for solving them, and some examples of common and useful equations. We introduce the main ideas in this chapter and describe them in a little more detail later in the course. Techniques for solving differential equations can take many different forms, including direct solution, use of graphs, or computer calculations. Solving such equations often provides information about how quantities change and frequently provides insight into how and why the changes occur. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y = f ( x ) y = f ( x ) and its derivative, known as a differential equation. 4.1.5 Identify whether a given function is a solution to a differential equation or an initial-value problem.Ĭalculus is the mathematics of change, and rates of change are expressed by derivatives.4.1.4 Identify an initial-value problem.4.1.3 Distinguish between the general solution and a particular solution of a differential equation.4.1.2 Explain what is meant by a solution to a differential equation.4.1.1 Identify the order of a differential equation.
#Diff eq full#
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