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homework set 3-Solutions

# homework set 3-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 # 3 U. Mu˘gan June 20, 2008 Homework problems from the 2 nd Edition, SECTION 1.5 5 (5) ) 2 Linear first order D.E. which can be solved by introducing the integrating factor ρ ( x ) ρ = exp •Z 2 x dx = e 2 ln x = x 2 . The D.E. can be written as d dx ( yx 2 ) = 3 x 2 Integration yields y ( x ) = x + c x 2 , c = constant I.C. implies that c = 4, so the solution of the I.V.P. is y ( x ) = x + 4 x 2 . 10 (10) ) Integrating factor is ρ ( x ) = exp •Z - 3 2 x dx = e ( - 3 ln x ) / 2 = x - 3 / 2 . Then the D.E. can be written as d dx ( yx - 3 / 2 ) = 9 2 x 1 / 2 Therefore the general solution of the D.E. is y ( x ) = 3 x 3 + cx 3 / 2 , c = constant . 14 (14) ) Integrating factor is ρ ( x ) = exp •Z - 3 x dx = e - 3 ln x = x - 3 . Then the D.E. can be written as d dx ( yx - 3 ) = 1 x 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

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Therefore the general solution of the D.E. is y ( x ) = x 3 ln x + cx 3 , c = constant . I.C. implies that c = 10, so the solution of the I.V.P. is y ( x ) = x 3 ln x + 10 x 3 . 20 (20) ) Integrating factor is ρ ( x ) = exp •Z ( - 1 - x ) dx = e - x - x 2 2 .
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homework set 3-Solutions - BILKENT UNIVERSITY Department of...

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