9.8HW-solutions - -x + h ( y ) y = y = x + h ( y )...

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MATH 2930: HW Solutions: Assigned September 8, 2010 Prepared by Alex Moore (problem #10 graded) Section 2.6 3.) (3 x 2 - 2 xy + 2)d x + (6 y 2 - x 2 + 3)d y = 0 ∂M ∂y = (3 x 2 - 2 xy + 2) ∂y = - 2 x ∂N ∂x = (6 y 2 - x 2 + 3) ∂x = - 2 x ∂M ∂y = ∂N ∂x This is an exact equation. We can use M to solve for the function: ψ = Z M d x = Z (3 x 2 - 2 xy + 2)d x = x 3 - x 2 y + 2 x + h ( y ) ψ y = ∂ψ ∂y = x 2 + h 0 ( y ) Comparing this to: N = (6 y 2 - x 2 + 3) h 0 ( y ) = 6 y 2 + 3 h ( y ) = 2 y 3 + 3 y + C The final answer becomes: x 3 - x 2 y + 2 x + 2 y 3 + 3 y = C 1
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10.) ( y/x + 6 x )d x + (ln x - 2)d y = 0 ∂M ∂y = ( y/x + 6 x ) ∂y = 1 /x ∂N ∂x = (ln x - 2) ∂x = 1 /x ∂M ∂y = ∂N ∂x This is an exact equation. We can use M to solve for the function: ψ = Z M d x = Z ( y/x + 6 x )d x = y ln x + 3 x 2 + h ( y ) ψ y = ∂ψ ∂y = ln x + h 0 ( y ) Comparing this to: N = ln x - 2 h 0 ( y ) = - 2 h ( y ) = - 2 y + C The final answer becomes: y ln x + 3 x 2 - 2 y = C 14.) Note that the equation is in the form: M ( x,y )d x - N ( x,y )d y = 0 with a minus sign instead of a plus. (9 x 2 + y - 1)d x - (4 y - x )d y = 0 ∂M ∂y = (9 x 2 + y - 1) ∂y = 1 ∂N ∂x = - (4 y - x ) ∂x = 1 2
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∂M ∂y = ∂N ∂x This is an exact equation. We can use M to solve for the function: ψ = Z M d x = Z (9 x 2 + y - 1)d x = 3 x 3 + xy
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Unformatted text preview: -x + h ( y ) y = y = x + h ( y ) Comparing this to: N =-4 y + x h ( y ) =-4 y h ( y ) = 2 y 2 + C The nal answer becomes: 3 x 3 + xy-x-2 y 2 = C 16.) ( ye 2 xy + x )d x + ( bxe 2 xy )d y = 0 M y = ( ye 2 xy ) y = e 2 xy + 2 xye 2 xy N x = ( bxe 2 xy ) y = be 2 xy + 2 bxye 2 xy Equation is exact if b = 1. Using M to solve (and integrating by parts): = Z M d x = Z ( ye 2 xy + x )d x = 1 2 e 2 xy + 1 2 x 2 + h ( y ) y = y = xe 2 xy + h ( y ) Comparing this to: N = xe 2 xy h ( y ) = 0 h ( y ) = C 3 The answer becomes: 1 2 e 2 xy + 1 2 x 2 = C We can also multiply both sides by 2 and change 2 C C to get the answer in the form: e 2 xy + x 2 = C 4...
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This note was uploaded on 01/21/2011 for the course MATH 2930 taught by Professor Terrell,r during the Spring '07 term at Cornell University (Engineering School).

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9.8HW-solutions - -x + h ( y ) y = y = x + h ( y )...

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