homework set 4-Solutions - BILKENT UNIVERSITY Department of...

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B ˙ ILKENT UNIVERSITY Department of Mathematics MATH 225, LINEAR ALGEBRA and DIFFERENTIAL EQUATIONS, Solution of Homework set 1 # 4 U. Mu˘gan June 20, 2008 Homework problems from the 2 nd Edition, SECTION 1.6 3 (3) ) 2 Let v = y/x then D.E. leads to following separable D.E. xv 0 = 2 v 1 / 2 then Z 1 2 v - 1 / 2 dv = Z dx x , v 1 / 2 = ln x + c, c = constant Therefore the solution is y ( x ) = x (ln x + c ) 2 . Note that the given D.E. is of homogenous type. 9 (9) ) If we let v = y/x then D.E. leads to following separable D.E. xv 0 = v 2 then Z dv v 2 = Z dx x , v - 1 = - ln x + c, c = constant Therefore the solution is x = y ( c - ln x ) . 10 (10) ) Let v = y/x then D.E. leads to following separable D.E. xvv 0 = 2 v 2 + 1 , Z 4 vdv 2 v 2 + 1 = Z 4 dx x ln(2 v 2 + 1) = c ln x + ln c, c = constant The solution of the given D.E. is x 2 + 2 y 2 = cx 6 . 12 (12) ) The given D.E. is of homogenous type, so let v = y/x , then D.E. leads to following separable D.E. xvv 0 = ( v 2 + 4) 1 / 2 1 I made every effort to avoid the calculation errors and/or typos while I prepared the solution set. You are responsible to check all the solutions and correct the errors if there is any. If you find any errors and/or misprints, please notify me. 2 The number in the parenthesis denotes the problem number in the International Edition of the textbook
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Z vdv ( v 2 + 4) 1 / 2 = Z dx x , ( v 2 + 4) 1 / 2 = ln x + c, c = constant Hence the solution of the given D.E. is
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homework set 4-Solutions - BILKENT UNIVERSITY Department of...

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