By A. R. Forsyth
Vintage 19th-century paintings one in all the best remedies of the subject. Differential equations of the 1st order, basic linear equations with consistent coefficients, integration in sequence, hypergeometric sequence, resolution via yes integrals, many different issues. Over 800 examples. Index.
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Extra resources for A Treatise on Differential Equations
X3 Ϫ 5x2 ϩ x Ϫ 5 120. x3 Ϫ x2 ϩ 3x Ϫ 3 66. ͑2x Ϫ 1͒͑x ϩ 3͒ ϩ 3͑x ϩ 3͒ 67. ͑u ϩ 2͒͑u Ϫ 2͒͑u 2 ϩ 4͒ In Exercises 121–152, completely factor the expression. 68. ͑x ϩ y͒͑x Ϫ y͒͑ 121. x 3 Ϫ 16x 122. 12x 2 Ϫ 48 123. x 3 Ϫ x 2 124. 6x 2 Ϫ 54 x2 ϩ y2 ͒ In Exercises 69–74, factor out the common factor. 69. 4x ϩ 16 70. 5y Ϫ 30 71. 2x 3 Ϫ 6x 72. 3z3 Ϫ 6z2 ϩ 9z 75. x 2 Ϫ 64 76. x 2 Ϫ 81 77. 48y2 Ϫ 27 78. 50 Ϫ 98z2 Ϫ 80. 81. ͑x Ϫ 1͒2 Ϫ 4 25 2 36 y Ϫ 49 82. 25 Ϫ ͑z ϩ 5͒2 84. x 2 ϩ 10x ϩ 25 1 85. x 2 ϩ x ϩ 4 4 4 86.
2 Exponents and Radicals 19 Example 12 Rationalizing a Denominator with Two Terms Rationalize the denominator of 2 . 3 ϩ Ί7 Solution 2 2 ϭ 3 ϩ Ί7 3 ϩ Ί7 ϭ ϭ и 3 Ϫ Ί7 3 Ϫ Ί7 Multiply numerator and denominator by conjugate of denominator. 2͑3 Ϫ Ί7 ͒ ͑3͒2 Ϫ ͑Ί7 ͒2 Find products. In denominator, ͑a ϩ b͒͑a Ϫ b͒ ϭ a 2 Ϫ ab ϩ ab Ϫ b 2 ϭ a 2 Ϫ b 2. 2͑3 Ϫ Ί7 ͒ ϭ 3 Ϫ Ί7 2 Simplify and divide out common factors. Now try Exercise 93. In calculus, sometimes it is necessary to rationalize the numerator of an expression.
It may sometimes be necessary to change the sign of a factor by factoring out ͑Ϫ1͒ to simplify a rational expression, as shown in Example 3. Example 3 Simplifying a Rational Expression Write 12 ϩ x Ϫ x2 in simplest form. 2x2 Ϫ 9x ϩ 4 Solution 12 ϩ x Ϫ x2 ͑4 Ϫ x͒͑3 ϩ x͒ ϭ 2x2 Ϫ 9x ϩ 4 ͑2x Ϫ 1͒͑x Ϫ 4͒ ϭ Factor completely. Ϫ ͑x Ϫ 4͒͑3 ϩ x͒ ͑2x Ϫ 1͒͑x Ϫ 4͒ ϭϪ 3ϩx , 2x Ϫ 1 x ͑4 Ϫ x͒ ϭ Ϫ ͑x Ϫ 4͒ 4 Divide out common factors. Now try Exercise 35. 1. Recall that to divide fractions you invert the divisor and multiply.