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Sec. 12] Linear Differential Equations with Constant Coefficients_351
2971*. Solve the equation
knowing its particular solution y l =
2972. Solve the equation
-^.
x*(\nx-l)y"-xy' 4-y = 0,
knowing its particular solution y =x.
}
By the method of variation of parameters, solve the
inhomogeneous linear equations.
2973. **(/" xy'
following
= 3;t a
.
2974*. x*y" + xy' y = x 2
.
2975. y'" -f (/'-sec A:.
Sec. 12. Linear Differential Equations of Second Order
with Constant Coefficients
t. Homogeneous equations. A second-order linear equation with constant
coeflicients p and q without the right side is of the form
If k l and fc 2 are roots of the characteristic equation
U)
.Q t (2)
then the general solution of equation (1) is \vritten in one of the following
three ways:
1) (/-CVV + C.e*-* if 6, and k z are real and /?, ^ k.\
2) y-eV(C, + CV) if *, = *,;
3) (/ -^(^cnspx-HC^mpA) if *,=a + p and * a -=a pi (p 76 0).
2. Inhomogeneous equations. The general solution of a linear inhomogeneous
differential equation
may
be written in the form of a sum:
y" + py' + w=fW (3)
where y is the general solution of the corresponding equation (I) without
ritfht side and determined from formulas (1) to (3), and Y is a particular
solution of the given equation (3).
The function Y may be found by the method of undetermined coefficients
in the following simple cases:
1. f (x) = e a *P n (x), where P n (x) is a polynomial of degree n.
If a is not a root of the characteristic equation (2), that is, (p (a) 96 0,
then we put Y e**Q n (x) where Q n (x) is a polynomial of degree n with
undetermined coeflicients.
If a is a root of the characteristic equation (2), that is,