document.write( "Question 1186674: The differential equation
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\n" ); document.write( "\n" ); document.write( "The term M(x,y)dx+N(x,y)dy becomes an exact differential if the left hand side above is divided by y3. Integrating that new equation, the solution of the differential equation is
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Algebra.Com's Answer #817651 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "Let's write it this way\r\n" );
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document.write( "He tells us to divide through by y³ to make it an exact differential\r\n" );
document.write( "equation.  So Let's multiply through by y^-3 (same thing!)\r\n" );
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document.write( "Get the sign between them +\r\n" );
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document.write( "So\r\n" );
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document.write( " and \r\n" );
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document.write( "Now let's show that the DE is exact by showing the partial derivatives\r\n" );
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document.write( "They are equal, so the differential equation is now exact.\r\n" );
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document.write( "So let's integrate both sides with respect to both variables.\r\n" );
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document.write( "Notice these 3 facts: \r\n" );
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document.write( "1. When we integrate something containing dx with respect to the other letter y,\r\n" );
document.write( "we get some function of the letter of its differential dx, the letter x, say u(x).\r\n" );
document.write( "2. When we integrate something containing dy with respect to the other letter x,\r\n" );
document.write( "we get some function of the letter of its differential dy, the letter y, say v(y).\r\n" );
document.write( "3. When we integrate 0 with respect to either letter, we get the same constant C.\r\n" );
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document.write( "Let's integrate with respect to x:\r\n" );
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document.write( "Let's integrate with respect to y:\r\n" );
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document.write( "We must have gotten exactly the same thing when we integrated with\r\n" );
document.write( "respect to x that we got when we integrated with respect to y\r\n" );
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document.write( "Therefore this equation,\r\n" );
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document.write( "must be identical with this equation\r\n" );
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document.write( "The equations will be identical if and only if\r\n" );
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document.write( " and \r\n" );
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document.write( "Therefore the general solution is\r\n" );
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document.write( "Edwin

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