1550078481-Ordinary_Differential_Equations__Roberts_

N-th Order Linear Differential Equations 199

EXERCISES 4.2

In the following exercises use POLYRTS or computer routines

available to you to calculate the roots of appropriate polynomial

equations.

1. In 1225, Leonardo Fibonacci showed in his text Flos that the equation

x^3 + 2x^2 + lOx = 20

has no solution of the form a + Vb where a and b are rational num-

bers. Then he obtained the following root by some undisclosed numeri-

cal method.

Find all solutions of the given cubic equation and compare with the

above solution.

2. Find the roots of the following equations which da Coi sent to Tartaglia
   for solution in 1530.

x^3 + 3x^2 = 5 and x^3 + 6x^2 + 8x = 1000

3. Find the roots of the following quartic equation which da Coi sent to
   Cardan for solution in 1540.

x^4 + 6x^2 + 36 = 60x

4. In his Ars Magna of 1545 Cardan demonstrated that x = 5 ± v=T5

were the solutions of x^2 + 40 = lOx. Compare values you obtain with

the values obtained by Cardan.

5. Listed below are some cubic equations which Cardan solved. Compare
   the values you obtain with the values obtained by Cardan.

Equation

x^3 + lOx = 6x^2 + 4

x^3 + 21x = 9x^2 + 5

x^3 + 26x = 12x^2 + 12

Roots

2, 2 ± V2

5, 2± Y3

2, 5 ± Vl9

6. In 1567, Nicolas Petri of Deventer calculated 1 + vf2 to be the positive
   root of the quartic equation

x^4 + 6x^3 = 6x^2 + 30x + 11

and neglected the negative roots. Find all the roots of this equation and

compare with Petri's single positive root.