Worksheet 6 Solutions - MA 2930, March 2, 2011 Worksheet 6...

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MA 2930, March 2, 2011 Worksheet 6 1. For each of the following equations check if it is exact. If it is, solve it. If it is not, can you think of the minimal ”repair” that’d make it exact and then solve that? (i) xdx ( x 2 + y 2 ) 3 / 2 + ydy ( x 2 + y 2 ) 3 / 2 = 0 (ii) ( e x sin y + 3 y ) dx - (3 x + e x cos y ) dy = 0 (i) Let’s check the exactness of the equation: M y = ∂y [ x ( x 2 + y 2 ) 3 / 2 ] = - 3 xy ( x 2 + y 2 ) 5 / 2 N x = ∂x [ y ( x 2 + y 2 ) 3 / 2 ] = - 3 xy ( x 2 + y 2 ) 5 / 2 Since M y = N x , the equation is exact. Therefore, there is a function φ ( x,y ) such that our equation is simply = 0. In the case, φ x = M and φ y = N . So, φ ( x,y ) = Z Mdx = Z x ( x 2 + y 2 ) 3 / 2 dx = - 1 ( x 2 + y 2 ) 1 / 2 + ξ ( y ) where ξ ( y ) is an unknown function of y , determined as follows: φ y = N y ( x 2 + y 2 ) 3 / 2 + ξ 0 ( y ) = y ( x 2 + y 2 ) 3 / 2 Therefore, ξ 0 ( y ) = 0, and so, ξ ( y ) = c , a constant which we can take to be zero. Thus, the solution to the equation is φ ( x,y ) = - 1 ( x 2 + y 2 ) 1 / 2 = a or, more simply ( x 2 + y 2 ) 1 / 2 = a , or even, x 2 + y 2 = a , where a is an arbitrary constant. (ii) Let’s check the exactness of the equation: M y = ∂y [ e x sin y + 3 y ] = e x cos y + 3 1
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N x = ∂x [ - (3 x + e x cos y )] = - 3 - e x cos y Since M y = - N x , the equation is not exact. However, if we made the minus sign a plus, it’d become exact. The resulting exact equation is ( e x sin y + 3 y ) dx + (3 x + e x
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This note was uploaded on 06/10/2011 for the course MATH 2930 taught by Professor Terrell,r during the Spring '07 term at Cornell University (Engineering School).

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Worksheet 6 Solutions - MA 2930, March 2, 2011 Worksheet 6...

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