By A. R. Forsyth

Vintage 19th-century paintings one among the best remedies of the subject. Differential equations of the 1st order, basic linear equations with consistent coefficients, integration in sequence, hypergeometric sequence, resolution by way of convinced integrals, many different issues. Over 800 examples. Index.

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**Additional resources for A Treatise on Differential Equations **

**Example text**

A z(z − a) 2 − + 2 y −4 + x+3 5 45. 38. x−3 5 − 4 y +2 46. 39. 16. (b − 3)(b + 3)(b2 + 9) 17. xy(x + y) 37. 1 y 40. 1 a 19. (t − 2)(t + 2t + 4) 4t + 1 3 + t2 t 3 v+1 + v(v − 2) v3 21. (n + 3)(n2 − 3n + 9) 42. w−1 2 − w3 w(w − 3) For Exercises 23–50, simplify the given expression as much as possible. 43. 1 x y − x−y y x 23. 4(2m + 3n) + 7m 44. 1 1 1 − y x−y x+y 3 2 20. (m − 2)(m + 2m + 4m + 8m + 16) 22. (y + 2)(y 4 − 2y 3 + 4y 2 − 8y + 16) 24. 3 2m + 4(n + 5p) + 6n − a 47. x−2 y z x+2 48. x−4 y+3 y−3 x+4 49.

5 · 3 · 2 + 6 · 4 For Exercises 5–22, expand the given expression. 6. (x + y − r )(z + w − t) 2 8. (3b + 5) 9. (2c − 7)2 2 10. (4a − 5) 11. (x + y + z)2 2 12. (x − 5y − 3z) 13. (x + 1)(x − 2)(x + 3) 14. (y − 2)(y − 3)(y + 5) 15. (a + 2)(a − 2)(a2 + 4) 1 x 1 z 2 18. a z(z − a) 2 − + 2 y −4 + x+3 5 45. 38. x−3 5 − 4 y +2 46. 39. 16. (b − 3)(b + 3)(b2 + 9) 17. xy(x + y) 37. 1 y 40. 1 a 19. (t − 2)(t + 2t + 4) 4t + 1 3 + t2 t 3 v+1 + v(v − 2) v3 21. (n + 3)(n2 − 3n + 9) 42. w−1 2 − w3 w(w − 3) For Exercises 23–50, simplify the given expression as much as possible.

3 5 2 2 115 76 1 2 1 3 3 0 1 2 3 3 1 12 7 2 257 101 3 Some rational numbers on the real line. We will use the intuitive notion that the line has no gaps and that every conceivable distance can be represented by a point on the line. With these concepts in mind, we call the line shown above the real line. We think of each point on the real line as corresponding to a real number. The undefined intuitive notions (such as “no gaps”) can be made precise using more advanced mathematics. In this book, we let our intuitive notions of the real line serve to define the system of real numbers.