New PDF release: A Text Book of Engineering Mathematics. Volume II

By Pandey, Rajesh.

ISBN-10: 1441657983

ISBN-13: 9781441657985

ISBN-10: 9380257120

ISBN-13: 9789380257129

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2 (~'Y) dy = 0 x 1] dx - 2 -Ydy 1 + -y22 + - 2 x x = 0 X or [1+ :, ]dx + [~: dx - 2 ~ dy ] which is an exact or [1 + x12 ] dx + d (-yix) = 0 using method I Integrating term by term, we get ~ + (- x- yix) = C, where C is constant of integration x2 -1 - y2 = Cx is the required solution. or Method IV. In the equation Mdx + Ndy = 0 If ~ M (aN _aM) is a function of y alone, say f(y), Ox 8y · . factor IS . e then the mtegrattng J f(y) dy Example 23. Solve (xy3 + y) dx + 2 (x2y2 + X+ y4) dy = 0 Solution.

1999) Solution. The given equation may be rewritten as e X / Y (1 _ ~) + (1 + e X / Y) Y or V e (l-v)+(l+e V ) (v+y . dx putting x = vy or dy = = 0 :~) =0 dv v + Ydy dv veV + v + ve V + (1 + ev) y dy or eV or dv (v + e v ) + (1+ e v ) y - = 0 dy or (1 + e V) dv + dy v+ e V y - dx dy Integrating, log (v + ev ) + log y = = = 0 log C, when C is an arbitrary constant or log {(v + e v ) y} = log C or (v + e v ) y or [~ + e X/Y]y = C, putting v = x/y or (x + y ex/y) = C = 0 C Equations Reducible to Homogeneous Form.

CP. = e- X [Cl cos 2x + C2 sin 2x], where Cl and C2 are arbitrary constants and P.. : here a =-1 1 -x =-e 8 :. The required solution is y = CP. I. e. y = e- X(Cl cos 2x + C2 sin 2x) + ? e- x 8 Example 2. Solve (D -1)2 (D2 + 1)2 Y = eX . Solution. Here the auxiliary equation is (m -1)2 (m2 + 1)2 = 0 or m = 1, 1, ± i, ±i :. CF. I. = 1 eX (D _1)2 (D2 + 1)2 1 1 eX (D - 1)2 (12 + 1)2 1 1 X -e (D _1)2 (2)2 1 1 X -e (D _1)2 4 = eX 1 1 (D + 1_1)2 4 43 A Textbook ofEn~neerin~ Mathematics Volume - II = eX ~ !

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A Text Book of Engineering Mathematics. Volume II by Pandey, Rajesh.


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