hw_1_22_23_09

# hw_1_22_23_09 - Ma 221 Homework Solutions Due Date January...

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Ma 221 Homework Solutions Due Date : January 22 - 23 , 2009 2 . 2pg . 46 # 1 ̄ , 4 , 5 , 7 , 9 ̄ , 1 ̄ 1 ̄ , 15 , 1 ̄ 7 ̄ , 19 , 2 ̄ 1 ̄ , 2 ̄ 2 ̄ , 23 , 25 ( Underlined problems are not handed in ) In problems 1, 4 and 5, determine whether the given differential equation is separable. 1) dy dx 2 y 3 y 4 dx dy dx 2 y 3 y 4 dx dy 2 y 3 y 4 dx therefore, this equation is separable. 4.) ds dt t ln s 2 t 8 t 2 ds dt 2 t 2 ln s 8 t 2 ds dt 2 t 2 ln s 4  ds ln s 4 2 t 2 dt therefore, this equation is separable. 5.) s 2 ds dt s 1 st Writing the equation in the form ds dt s 1 st s 2 shows that the equation is not separable. In problems 7, 9, 11 and 15, solve the equation. 7.) dy dx 1 x 2 y 2 dy dx 1 x 2 y 2 y 2 dy 1 x 2 dx y 2 dy 1 x 2 dx 1 3 y 3 y 3 3 x x 3 C 9) dy dx y 2 sin x dy dx y 2 sin x   dy y 2 sin x dx dy y 2 sin x dx y e 2 x cos x C Ce 2 x cos x 11) dy dx sec 2 y 1 x 2 dy dx sec 2 y 1 x 2 dy sec 2 y dx 1 x 2 Using trigonemtric identities we have: 1

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sec y 1 cos y and cos 2 y 1 2 1 cos2 y dy sec 2 y dx 1 x 2 1 y dy 2 dx 1 x 2 1 y dy 2 dx 1 x 2 1 2 y 1 2 sin2 y arctan x C 1 2 y y 4arctan x 4 C 1 2 y y x C 15.) y 1 dy ye cos x sin xdx 0 ye cos x sin xdx y 1 dy e cos x sin xdx y 2 dy Substituting: let u cos x du sin xdx du sin xdx e u du y 2 dy e u y 1 C e cos x y 1 C y 1 C e cos x In problems 17, 19, 21, 22, 23, and 25, solve the initial value problem. 17) y x 3 1 y , y 0 3 dy dx x 3 1 y   dy 1 y x 3 dx dy 1 y x 3 dx ln|1 y | |1 y | exp C 1 x 4 4 Ce x 4 4 Substituting the IC y 0 3 |1 3| Ce 0 4 4 | 2| C 2 |1 y | 2 e x 4 4 Since 1 y 0 1 3 0, on an interval containing x 0 one has 1 y x 0 and so |1 y x | y x 1 . The solution is then: 2
y 1 2 e x 4 4 y 2 e x 4 4 1 19.) dy d y sin , y 3 dy d y sin dy y sin d dy y sin d ln| y | cos C 1 | y | e cos C 1 y Ce cos because at the initial point, , y 3 0 3 y Ce cos Ce 1 C 3 e 1 y 3 e 1 e cos 3 e 1 cos 21) dy dx 2 y 1 cos x , y 0 dy 2 y 1 cos

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## This note was uploaded on 09/22/2009 for the course MA MA221 taught by Professor Levine during the Spring '09 term at Stevens.

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hw_1_22_23_09 - Ma 221 Homework Solutions Due Date January...

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