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Engineering Mathematics

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Published in: Mathematics
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Definitions and Solutions of differential equations!

Hari B / Hyderabad

9 years of teaching experience

Qualification: M.Sc (jntuh campus - 2012)

Teaches: Algebra, Business Mathematics, Mathematics, Statistics, B.Tech Tuition, Polytechnic

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  1. 1. 2. 3. 4. 5. 6. 7. 8. (x — —b)(x —c) 9. an -bn 10. -b 6. PARTIAL FRACTIONS Quick Review An expression of the form f(x) = ao + avx + a2X +... + anxn, where n is a non negative integer and .. .,an are real numbers such that an 0, is called a polynomial in x of degree n. ao, al, a2, Division Algorithm : If f(x), 0 g(x) are two polynomials, then 3 polynomials q(x), r(x) uniquely such that f(x) = q(x) g(x) + r(x) where r(x) = 0 or deg r(x) < deg g(x). The polynomial q(x) is called quotient and the polynomial r(x) is called remainder of f(x) when divided by g(x). f(x) If f(x), g(x) are two polynomials and g(x) 0 then is called a rational function or rational fraction. f(x) A fraction is said to be a proper fraction if deg f(x) < deg g(x). Otherwise it is said to be an improper fraction. If a rational function can be expressed as a sum of two or more proper fractions, then each fraction is called a partial fraction of the given function. f(x) Let — be a proper fraction. (i) If (ax + where n e N, is a factor of g(x) then the partial fractions corresponding to this factor are where Al, A2,. .. An are constants. ax + b (ax + (ax + (ii) If (ax + bx + where n e N, is a factor of g(x) then the partial fractions corresponding to AIX this factor are are constants. pb + q pa + q (x —a)(a —b) (b — —b) Anx -k Bn where Al, A2,. .. An BI,B2,..., n pb + q (b — a)(x — b)(b —c) pc + q (c —a)(c —b)(x —c) x2(x —a) q -q 2 ax pa + q (x - -b)(a -c) pa + q pa + q a2x a2(x -a) If IXI < a, IXI < b then the coefficient of xn In x (x — a)(x — b) 1 is a 1 anbn

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