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CivE222_Samples_Midterm

# CivE222_Samples_Midterm - University of Waterloo Department...

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University of Waterloo Department of Civil and Environmental Engineering CivE 222: Differential Equations Midterm Examination Date: 1:30 pm 3:30 pm , Monday, June 16, 2008 Closed Book Instructor: Prof. Wei-Chau Xie Spring 2008 Important Notes: One page (letter size, one-sided) of formula sheet is allowed. Tables of Trigonometric Identities, Derivatives, Integrals handed out at the beginning of the term are allowed. Each question carries the same mark. Solve any six of the following eight differential equations. If you attempt more than six questions, the best six marks will be taken. If you have time, try the bonus queations, which are added on top of your total marks. 1 . y + y x = e xy Answer: e x y = 1 2 x 2 + C 2 . yy ′′ yy = ( y ) 2 Answer: ln vextendsingle vextendsingle vextendsingle ln vextendsingle vextendsingle y vextendsingle vextendsingle + C vextendsingle vextendsingle vextendsingle = x + C 1 ; y = C 3 . y = 1 xy + x 3 y 3 Answer: 1 x 2 = 1 y 2 + C e y 2 4 . ( e x + 3 y 2 ) d x + 2 xy d y = 0 Answer: e x ( x 2 2 x + 2 ) + x 3 y 2 = C 5 . ( xy + 2 x 3 y ) d x + x 2 d y = 0 Answer: ln vextendsingle vextendsingle xy vextendsingle vextendsingle + x 2 = C or xy e x 2 = C 6 . x ( y ) 2 2 yy + 4 x = 0 Answer: y = C 2 x 2 + 2 C ; y = ± 2 x 7 . y ′′′ = 2 ( y ′′ 1 ) cot x Answer: y = parenleftBig C 2 2 + 1 parenrightBig x 2 2 + C 2 8 cos 2 x + C 1 x + C 0 8 . xy = y + radicalBig x 2 y 2 , x > 0 Answer: sin 1 y x = ln vextendsingle vextendsingle x vextendsingle vextendsingle + C ; y = ± x Bonus.

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CivE222_Samples_Midterm - University of Waterloo Department...

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