document.write( "Question 637388: Solve the differential equation:\r
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Algebra.Com's Answer #401709 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"3e%5Ex%2Atan%28y%29dx\" + \"%281-e%5Ex%29%2Asec%5E2%28y%29dy\" = 0\r\n" );
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document.write( "Separate the variables by dividing through by \"%281-e%5Ex%29tan%28y%29%0D%0A\"\r\n" );
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document.write( "\"%283e%5Ex%2Atan%28y%29dx%29%2F%28%281-e%5Ex%29tan%28y%29%29\" + \"%28%281-e%5Ex%29%2Asec%5E2%28y%29dy%29%2F%28%281-e%5Ex%29tan%28y%29%29\" = \"0%2F%28%281-e%5Ex%29tan%28y%29%29\"\r\n" );
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document.write( "\"%283e%5Ex%2Across%28tan%28y%29%29dx%29%2F%28%281-e%5Ex%29cross%28tan%28y%29%29%29\" + \"%28%28cross%281-e%5Ex%29%29%2Asec%5E2%28y%29dy%29%2F%28%28cross%281-e%5Ex%29%29tan%28y%29%29\" = 0\r\n" );
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document.write( "\"%283e%5Ex%2Adx%29%2F%281-e%5Ex%29\" + \"%28sec%5E2%28y%29dy%29%2Ftan%28y%29\" = 0\r\n" );
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document.write( "Now we integrate all three terms:\r\n" );
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document.write( "\"int%28%283e%5Ex%29dx%2F%281-e%5Ex%29%29%29\" + \"int%28%28sec%5E2%28y%29dy%29%2Ftan%28y%29%29\"= \"int%280%29\"\r\n" );
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document.write( "Both terms on the left can be integrated using the formula \"int%28du%2Fu%29\" = ln|u|+C\r\n" );
document.write( "The second integral is already set up for that. We take the 3 out\r\n" );
document.write( "of the integral on the first term:\r\n" );
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document.write( "\"3int%28%28e%5Ex%29dx%2F%281-e%5Ex%29%29%29\" + \"int%28%28sec%5E2%28y%29dy%29%2Ftan%28y%29%29\"= \"int%280%29\"\r\n" );
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document.write( "We need a negative sign in the numerator of the first fraction so it will\r\n" );
document.write( "be in the form:\r\n" );
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document.write( "\"-3int%28%28-e%5Ex%29dx%2F%281-e%5Ex%29%29%29\" + \"int%28%28sec%5E2%28y%29dy%29%2Ftan%28y%29%29\"= \"int%280%29\"\r\n" );
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document.write( "Integrating using ln(C) for the arbitrary constant so everything will\r\n" );
document.write( "be natural logs:\r\n" );
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document.write( "-3·ln|1-ex| + ln|tan(y)| = ln(C)\r\n" );
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document.write( "               ln|tan(y)| = 3·ln|1-ex| + ln(C)\r\n" );
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document.write( "Use a rule of logs to move the 3 to an exponent:\r\n" );
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document.write( "               ln|tan(y)| = ln|1-ex|3 + ln(C)\r\n" );
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document.write( "Write the right side as the natural log of a product:\r\n" );
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document.write( "               ln|tan(y)| = ln[C|1-ex|3]\r\n" );
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document.write( "Take anti-logs of both sides:\r\n" );
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document.write( "               |tan(y)| = C|1-ex|3\r\n" );
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document.write( "Since the constant C can be positive or negative, we don't need\r\n" );
document.write( "the absolute values:\r\n" );
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document.write( "               tan(y) = C(1-ex)3\r\n" );
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document.write( "And if we like we can solve for y by taking arctangents of both sides:\r\n" );
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document.write( "                    y = arctan[C(1-ex)3]\r\n" );
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document.write( "Edwin
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