document.write( "Question 1199306: Question\r
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Algebra.Com's Answer #833194 by Edwin McCravy(20054) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "That is easily solvable by separating the variables.  Its solution is y = Cx\r\n" );
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document.write( "But your professor wants you to tackle it as either an exact differential \r\n" );
document.write( "equation or one solvable by using an integrating factor to convert it to\r\n" );
document.write( "an exact differential equation. \r\n" );
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document.write( "First we check to see if the differential equation is exact:\r\n" );
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document.write( "We write it in the form \r\n" );
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document.write( "So M(x,y) = y,   N(x,y) = -x\r\n" );
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document.write( "We form the partial derivatives:\r\n" );
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document.write( "Those are not equal, so the differential equation is not exact.\r\n" );
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document.write( "So we can try one of these integrating factors to multiply through by to\r\n" );
document.write( "see if it comes out to be an exact differential equation:\r\n" );
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document.write( "Trying the first one:\r\n" );
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document.write( "So we multiply the original differential equation\r\n" );
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document.write( " through by \r\n" );
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document.write( "Let's see if that is exact.\r\n" );
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document.write( "Taking the partial of  with respect to y givss \r\n" );
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document.write( "Taking the partial of  with respect to x \r\n" );
document.write( "gives \r\n" );
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document.write( "They are not equal so this integrating factor did not work.  There is no\r\n" );
document.write( "use to try the other one because the equation is symmetrical in x and y,\r\n" );
document.write( "and it would be the same with the x's and y's swapped.\r\n" );
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document.write( "So we can't convert the given differential equation to an exact one by\r\n" );
document.write( "the usual method.  That's because the method assumes it's possible to\r\n" );
document.write( "find an integrating factor in terms of one variable only.  So it's not\r\n" );
document.write( "possible.  So we can't do what your professor asked you to do.\r\n" );
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document.write( "So let's just solve it by separating the variables:\r\n" );
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document.write( "I like my dy to be at the first:\r\n" );
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document.write( "Divide both sides by \r\n" );
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document.write( "Edwin

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