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

By Pandey, Rajesh.

ISBN-10: 9380257120

ISBN-13: 9789380257129

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Extra info for A Text Book of Engineering Mathematics. Volume II

Example text

X dx _ ! z + log x -log Y = C, where C is an arbitrary constant ~ + log x -log Y = C, putting z or - or 1 log (xjy) = C + xy xy = xy Example 19. S. 1993) . Solution. The given equation can be rewritten as 26 Differential Equations o(First Order and First Dei{'ee xy2 (1 + 2xy) dx + x2y (1 - xy) dy = 0 or Y (1+ 2xy) dx + x (1 - xy) dy = 0 or (y dx + x dy) + 2xy2 dx - x2y dy = 0 or d (xy) + 2xy2 dx - x2y dy = 0 Dividing both sides of this equation by x2y2, we get d (xy) 2 1 - + - dx - - dy = 0 2 x y2 X Y or (:2) dz+ (z/x) dx- (1/y) dy=O, wherez=xy Integrating, - .!.

Solution of a exact differential equation is fMdx fNdy =c + Regarding yasa constant only those termsofN not containing x Example 16. S. 1993) Solution. Here M = Y (1 + :. aM ay = aM ay =:> - (1 1). + -; - sm y an aN = -, ax ~) cos y and N = x + log x - x sin Y daNax =1 1 . y + -; - sm h ence t h · " IS exact e gIven equation Regarding y as constant, fMdx = f{y R1 (1 + ;) + cos y} dx ~) dx + cos Y fdx = y = Y(x + log x) + (cos y) x + (1) Also no new term is obtained by integrating N with respect to y.

S. 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.

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