document.write( "Question 474425: Solve the differential equation (x^2 - 8x + 15) dy/dx = 6y + 5 \n" ); document.write( "
Algebra.Com's Answer #325391 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\"%28x%5E2+-+8x+%2B+15%29expr%28dy%2Fdx%29+=+6y+%2B+5\"
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document.write( "Multiply through by dx\r\n" );
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document.write( "\"%28x%5E2+-+8x+%2B+15%29dy+=+%286y+%2B+5%29dx\"\r\n" );
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document.write( "Separate the variables:\r\n" );
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document.write( "\"dy%2F%286y%2B5%29+=+dx%2F%28x%5E2-8x%2B15%29\"\r\n" );
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document.write( "Integrating the left side using formula \"int%28du%2Fu%29\" = \"ln%28abs%28u%29%29%2BC\"\r\n" );
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document.write( "\"int%28dy%2F%286y%2B5%29%29\" = \"expr%281%2Fred%286%29%29int%28red%286%29dy%2F%286y%2B5%29%29\" = \"expr%281%2F6%29ln%286y%2B5%29%2BC%5B1%5D\"\r\n" );
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document.write( "To integerate the right side, we must first complete the\r\n" );
document.write( "square on the denominator:\r\n" );
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document.write( "x² - 8x + 15\r\n" );
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document.write( "multiply -8 by 1/2, get -4, then square, getting +16\r\n" );
document.write( "Add 16 then subtract 16\r\n" );
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document.write( "x² - 8x + 16 - 16 + 15\r\n" );
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document.write( "Factor first three terms as square of a binomian, combine last two terms:\r\n" );
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document.write( "(x - 4)² - 1\r\n" );
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document.write( "\"int%28dx%2F%28x%5E2-8x%2B15%29%29\" = \"int%28dx%2F%28%28x-4%29%5E2-1%29%29\"\r\n" );
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document.write( "We use formula \"int%28du%2F%28u%5E2-a%5E2%29%29\" = \"expr%281%2F%282a%29%29ln%28abs%28%28u-a%29%2F%28u%2Ba%29%29%29%2BC\"\r\n" );
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document.write( "\"int%28dx%2F%28%28x-4%29%5E2-1%29%29\" = \"expr%281%2F%282%2A1%29%29ln%28abs%28%28%28x-4%29-1%29%2F%28%28x-4%29%2B1%29%29%29%2BC%5B2%5D\" = \"expr%281%2F2%29ln%28abs%28%28x-5%29%2F%28x-3%29%29%29%2BC%5B2%5D\"\r\n" );
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document.write( "So we have:\r\n" );
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document.write( "\"expr%281%2F6%29ln%286y%2B5%29%2BC%5B1%5D\" = \"expr%281%2F2%29ln%28abs%28%28x-5%29%2F%28x-3%29%29%29%2BC%5B2%5D\"\r\n" );
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document.write( "Let arbitrary constant \"C%5B3%5D+=+C%5B2%5D-C%5B1%5D\"\r\n" );
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document.write( "\"expr%281%2F6%29ln%286y%2B5%29\" = \"expr%281%2F2%29ln%28abs%28%28x-5%29%2F%28x-3%29%29%29%2BC%5B3%5D\"\r\n" );
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document.write( "Multiply through by 6\r\n" );
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document.write( "\"ln%286y%2B5%29\" = \"3ln%28abs%28%28x-5%29%2F%28x-3%29%29%29%2B3C%5B3%5D\"\r\n" );
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document.write( "Use the coefficient/exponent rule of logarithms and \r\n" );
document.write( "write arbitrary constant \"3C%5B3%5D\" as \"ln%28C%29\",\r\n" );
document.write( "choosing C > 0.\r\n" );
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document.write( "\"ln%286y%2B5%29\" = \"ln%28abs%28%28x-5%29%2F%28x-3%29%29%5E3%29%2Bln%28C%29\"\r\n" );
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document.write( "Use the sum/product rule of logarithms on the right:\r\n" );
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document.write( "\"ln%286y%2B5%29\" = \"ln%28C%2Aabs%28%28x-5%29%2F%28x-3%29%29%5E3%29\"\r\n" );
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document.write( "\"6y%2B5\" = \"C%2Aabs%28%28x-5%29%2F%28x-3%29%29%5E3\"\r\n" );
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document.write( "If the arbitrary constant is chosen properly we may \r\n" );
document.write( "erase the absolute values:\r\n" );
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document.write( "\"6y%2B5\" = \"C%2Aexpr%28%28x-5%29%2F%28x-3%29%5E3%29\"\r\n" );
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document.write( "or if we like:\r\n" );
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document.write( "\"%286y%2B5%29%28x-3%29%5E3%29\" = \"C%28x-5%29\"\r\n" );
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document.write( "Edwin

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