高等数学英文版课件PPT 15 Differential equations.ppt

Differential equations
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First-order linear equations
Exact equations
Strategy for solving first-order equations
Chapter 15
Basic concepts, separable and homogeneous equations
A second-order linear differential equation has the form
(1)
where P, Q, R, and G are continuous functions.
Second-Order Linear Equations
If G(x) = 0 for all x, such equations are called second-order homogeneous linear equations. (This use of the word homogeneous has nothing to do with the meaning given in Section .)
(2)
If for some x, Equation 1 is nonhomogeneous.
Two basic facts enable us to solve homogeneous linear equations. The first of these says that if we know two solutions and of such an equation, then the bination is also a solution.
(3)Theorem If and are both solutions of the linear equation (2) and and are any constants, then the function is also a solution of Equation 2.
Proof Since and are solutions of Equation 2, we have
and
Therefore
Thus is a solution of Equation 2.
Let x and y are two variables, if neither x nor y is a constant multiple of the other, we say x and y are two linearly independent variables. For instance, the function
and are linearly dependent, but
and are linearly independent.
The second theorem says that the general solution of a homogeneous linear equation is a bination of two linearly independent solutions.
(4)Theorem If and are linearly independent solutions of Equation 2 , then the general solution is given by
where and are arbitrary constants.
In general, it is not easy to discover particular solutions to a second-order linear equation. But it is always possible to do so if the coefficient functions P, Q and R are constant functions, that is, if the differential equation has the form
(5)
It is not hard to think of some likely candidates for particular solutions of Equation 5. For example, the exponential function y because its derivatives are constants multiple of itself: . Substitute the