Assignment 1 Correction

# Assignment 1 Correction - MATH-315 2011(Winter(Ordinary...

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Unformatted text preview: MATH-315 2011 (Winter) (Ordinary Differential Equations) Solutions to Written-Assignment #1 February 5, 2011 Solution of 1(a): One can rewrite the given equation as y ( y 2- 1) − 4 / 3 dy = xdx, which is a separable equation, so that a simple integration yields ( y 2- 1) − 1 / 3 =- 1 / 3( x 2 + c ) . The initial condition determines c as c =- 3( b 2- 1) − 1 / 3 . Therefore y = √ 1 + (- 1 3 x 2 + ( b 2- 1) − 1 / 3 ) − 3 . Note that we have to choose the + sign since b > 0. In order to find the interval of definition, we need to know for which values of x , the expression under the radical is non-negative. That is, when is (- 1 3 x 2 + ( b 2- 1) − 1 / 3 ) − 3 ≥ - 1? To this end, we first make the following simple observation. For A ∈ R- { } , A − 3 ≥ - 1 ⇐⇒ A > or A ≤ - 1 . Now we consider two cases: (I) b > 1. Then obviously b 2- 1 > 0. With A = [- 1 3 x 2 + ( b 2- 1) − 1 / 3 ] , note that A > ⇐⇒ | x | < √ 3 ( b 2- 1) 1 / 3 ; and that A ≤ - 1 ⇐⇒ | x | ≥ √ 3 + 3 ( b 2- 1) 1 / 3 . 1 Therefore, in this case, the interval of definition is { x ∈ R : | x | < √ 3 ( b 2- 1) 1 / 3 } ∪ { x ∈ R : | x | ≥ √ 3 + 3 ( b 2- 1) 1 / 3 } . (II) 0 < b < 1. Then obviously, 0 < 1- b 2 < 1 . We can write A = [- 1 3 x 2- | b 2- 1 | − 1 / 3 ] . In this case, A > 0 cannot happen(why?)....
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Assignment 1 Correction - MATH-315 2011(Winter(Ordinary...

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