188 Ordinary Differential Equations
17. a. Verify that YP = 3x + 2 is a particula r solution on (-oo, oo) of the
nonhomogeneous linear differential equation(31) y" + 9y = 27x + 18.
b. Write the associated homogeneo us equation.c. Verify that y 1 = sin 3x and Y2 = cos 3x are linearly indep endent
solutions on ( -oo, oo) of t he associated homogeneous equation.
d. Write the complementary solution for (31).
e. Write the general solution of (31).
f. Find the solution of (31) which satisfies the initial conditions
y(O) = 23, y' (0) = 21.18. a. Verify that YP = x + 1/x is a p a rticular solution on (0, oo) of the
nonhomogeneous, linear differential equation(32) x^2 y" + xy' - 4y = -3x - 3/x.
b. Write the associated homogeneous equation.c. Verify that y 1 = x^2 and Y2 = 1 /x^2 are linearly independent solutions
on (0, oo) of the associated homogeneous equation.
d. Write the complem entary solution for (32).
e. Write the general solution of (32).
f. Find the solution of (32) which satisfies the initial conditions
y(l) = 3, y'(l) = -6.4.2 Roots of Polynomials
Previously you found roots of polynomials in order to solve word problems
in algebra; to aid in the graphing of polynomials; and to find critical points,
relative maxima, and relative minima of polynomials in calculus. In the next
section, we will see how roots of polynomials enter into the solut ion of dif-
ferential equations. Furthermore, as you continue to study mathematics you
will encounter a multitude of other occasions on which you will need to find
the roots of a polynomial.