(5)
It is not hard to think of some likely candidates for particular solutions of Equation 5. For example, the exponential function y erx because its derivatives are constants multiple of itself: y rerx, y r 2erx . Substitute these expression into Equation 5
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15.5 Second-Order Linear Equations
A second-order linear differential equation has the form
(1) P(x) d 2 y Q(x) dy R(x) y G(x)
dx 2
dx
is also a solution of Equation 2.
Thus
is a solution of Equation 2.
y c1y1 [P(x) y1 Q(x) y1 R(x) y1] c2[P(x) y2 Q(x) y2 R(x) y2 ] P(x)(c1 y1 c2 y2) Q(x)(c1 y1 c2 y2 ) R(x)(c1 y1 c2 y2 ) P(x)(c1 y1 c2 y2 ) Q(x)(c1 y1 c2 y2 ) R(x)(c1 y1 c2 y2 )
(4)Theorem If y1 and y2 are linearly independent
solutions of Equation 2 , then the general solution is given by
where c1 and c2 are arbitrary constants.
f (x) x2 and g(x) 5x2 are linearly dependent, but f (x) e x and g(x) xe x are linearly independent.
The second theorem says that the general solution of a homogeneous linear equation is a linear combination of two linearly independent solutions.
y(x) c1y1(x) c2 y2 (x)
ay by cy 0
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
Differential equations
Chapter 15
15.1 Basic concepts, separable and homogeneous equations
15.2 First-order linear equations 15.3 Exact equations 15.4 Strategy for solving first-order equations
where P, Q, R, and G are continuous functions.
If G(x) = 0 for all x, such equations are called secondorder homogeneous linear equations. (This use of the word homogeneous has nothing to do with the meaning given in Section 15.1.)
solutions y1 and y2 of such an equation, then the linear
combination
is also a solution.
y(x) c1y1(x) c2 y2 (x) y c1y1 c2 y2
(3)Theorem If y1(x) and y2 (x) are both solutions of the linear equation (2) and c1 and c2 are any constants, then the function
(2) P(x) d 2 y Q(x) dy R(x) y 0
dx 2
dx
If G(x) 0 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
P(x) y Q(x) y R(x) y
P(x) y2 Q(x) y2 R(x) y2 0
P(x) y1 Q(x) y1 R(x) y1 0
Proof Since y1 and y2 are solutions of Equation 2, we have and
Therefore
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