By A. R. Forsyth

ISBN-10: 0486693147

ISBN-13: 9780486693149

Vintage 19th-century paintings one of the best remedies of the subject. Differential equations of the 1st order, normal linear equations with consistent coefficients, integration in sequence, hypergeometric sequence, answer through convinced integrals, many different themes. Over 800 examples. Index.

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

**Example text**

To paraphrase an old saying, there are three components to success—simplify, simplify, simplify. The easiest two-digit addition problems are those that do not require you to carry any numbers, when the first digits sum to 9 or below and the last digits sum to 9 or below. For example: 47 ؉ 32 (30 ؉ 2) ᎏᎏ To solve 47 ؉ 32, first add 30, then add 2. After adding 30, you have the simpler problem 77 ؉ 2, which equals 79. We illustrate this as follows: 47 ؉ 32 ؍ 77 ؉ 2 (first add 30) ؍ 79 (then add 2) The above diagram is simply a way of representing the mental processes involved in arriving at an answer using our method.

53 ؋ 5 ᎏ 9. 84 ؋ 5 ᎏ 10. 58 ؋ 6 ᎏ 11. 97 ؋ 4 ᎏ 12. 78 ؋ 2 ᎏ 13. 96 ؋ 9 ᎏ 14. 75 ؋ 4 ᎏ 15. 57 ؋ 7 ᎏ 16. 37 ؋ 6 ᎏ 17. 46 ؋ 2 ᎏ 18. 76 ؋ 8 ᎏ 19. 29 ؋ 3 ᎏ 20. qxd 5/4/06 1:45 PM Page 36 36 Secrets of Mental Math 3-BY-1 MULTIPLICATION PROBLEMS Now that you know how to do 2-by-1 multiplication problems in your head, you will find that multiplying three digits by a single digit is not much more difficult. You can get started with the following 3-by-1 problem (which is really just a 2-by-1 problem in disguise): 320 (300 ؉ 20) ؋ ᎏ ᎏᎏᎏᎏ7 300 ؋ 7 ؍2100 20 ؋ 7 ؍؉ᎏ1 ᎏ0 ᎏ ᎏ ᎏ4 2240 Was that easy for you?

If, after you’ve worked out these problems, you would like to practice more, make up your own. Calculate mentally, then check your answer with a calculator. Once you feel confident that you can perform these problems rapidly in your head, you are ready to move to the next level of mental calculation. EXERCISE: 2-BY-1 MULTIPLICATION 1. 82 ؋ 9 ᎏ 2. 43 ؋ 7 ᎏ 3. 67 ؋ 5 ᎏ 4. 71 ؋ 3 ᎏ 5. 93 ؋ 8 ᎏ 6. 49 ؋ 9 ᎏ 7. 28 ؋ 4 ᎏ 8. 53 ؋ 5 ᎏ 9. 84 ؋ 5 ᎏ 10. 58 ؋ 6 ᎏ 11. 97 ؋ 4 ᎏ 12. 78 ؋ 2 ᎏ 13. 96 ؋ 9 ᎏ 14. 75 ؋ 4 ᎏ 15.

### A Treatise on Differential Equations by A. R. Forsyth

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