Final solutions

# Final solutions - MCGILL UNIVERSITY SOLUTIONS OF FINAL EXAM...

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MCGILL UNIVERSITY SOLUTIONS OF FINAL EXAM OF MATH-315, ORDINARY DIFFERENTIAL EQUATIONS (APRIL 13, 2011, 9:00AM) (THE FULL MARK IS 100 POINTS ) (1) ( 8 points ) Find general solution for EQ: x 3 y 0 + 2 xy - y 3 = 0 , ( x > 0) . Solution: This is a Bernoulli Eq, whose general solution can be derived as y ( x ) = ± 5 x (2 + 5 Cx 5 ) 1 / 2 . (2) ( 8 points ) Given EQ: 1 + 2 x y 2 + 2 x y 2 y 0 = 0 . Solution: (a) Check whether or not it is exact EQ; Not exact, as M y - N x = - 4 x y 3 - 2 y 2 6 = 0. (b) The problem does not have an integral factor. The students, as long as tried the the cases of integrating factor in the forms: μ ( y ); μ ( x ) or P ( x ) Q ( y ) and showed no such kinds of integrating factors, will get the full points. (3) ( 12 points ) By using the method of differential operators, solve y 00 + 2 y 0 + y = 2e - x + sin x. (a) Determine what is the annihilator of the inhomogeneous term; (b) Find a particular solution; (c) Write the general solution for the equation. Solution: P ( D ) = ( D + 1) 2 Q 1 ( D ) = ( D + 1); Q 2 ( D ) = ( D 2 + 1) .

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