210 Ordinary Differential Equations
So the real, general solution to the given differential equation is
y = c 1 e-x + c2e^3 x cos 2x + C3e^3 x sin 2x + C4xe^3 x cos 2x + csxe^3 x sin 2x
= cie-x + (c2 + c4x)e^3 x cos 2x + (c3 + C5X )e^3 x sin 2x.EXERCISES 4.3
In exercises 1-10 determine the real general solution of the given
homogeneous linear differential equations.
1. 4y(^2 ) + 4y(l) - 3y = 0
- yC^3 ) - 4yC^2 ) + 6y(l) - 4y = 0
- yC^4 ) - l6y = 0
- yC^4 ) + l6y = 0
- yC^4 ) - 4yC^3 ) + 8yC^2 l - 8y(l) + 4y = 0
6. yC^4 ) - 8y(l) = 0
- 36yC^4 ) - 12yC^3 ) - lly(^2 ) + 2y(l) + y = 0
- yC^5 ) - 3y(^4 ) + 3yC^3 ) - 3yC^2 l + 2y(l) = 0
- yC^5 ) - yC^4 ) + yC^3 ) + 35y(^2 ) + 16y(l) - 52y = 0
- yC^8 ) + 8y(^4 ) + l6y = 0
In exercises 11-15 assume a and /3 fa are real constants.
- Find the real general solution of y" + ay = 0 on (-oo, oo) for a. a > 0,
b. a = 0, and c. a < 0.
- Find the real general solution of y" + ay' = 0 on ( -oo, oo) for a. a f 0
and b. a = 0.
13. Find the real general solution of y" - 2ay' + a^2 y = 0 on ( -oo, oo) for
af 0.- Find the real general solution of y" - (a+f3)y' +af3y = 0 on (-oo, oo).
- Find the real general solution of y" -2ay' + ( a^2 + /3^2 )y = 0 on ( -oo, oo ).
- Let a be a real constant and consider the differential equation
(*) y" + 2ay' + 1 = 0.
a. Write the auxiliary equation associated with (*).
b. Use the quadratic formula to find the roots of the auxiliary equa-
tion.