document.write( "Question 460471: The order of the differential equation of all parabolas having their axes of symmetry coinciding with x-axis is : \n" ); document.write( "
Algebra.Com's Answer #315831 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The answer is order 2.  Here is why:\r\n" );
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document.write( "First of all, the equation of a parabola with any horizontal \r\n" );
document.write( "axis of symmetry y = k is\r\n" );
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document.write( "(y-k)² = 4p(x-h)\r\n" );
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document.write( "That has three arbitrary constants h, k and p.\r\n" );
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document.write( "Therefore a differential equation of all parabolas \r\n" );
document.write( "having ANY horizontal axis of symmetry (not necessarily the\r\n" );
document.write( "x-axis, would be of order 3, the number of arbitrary constants.\r\n" );
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document.write( "If we restrict the family of parabolas to those with axis of \r\n" );
document.write( "symmetry y = 0, (where k = 0), the equations would be\r\n" );
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document.write( "y² = 4p(x-h)\r\n" );
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document.write( "y² = 4px - 4ph\r\n" );
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document.write( "or \r\n" );
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document.write( "y² = c1x + c2, where c1 = 4p and c2 = 4ph\r\n" );
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document.write( "That has 2 arbitrary constants, so the order would be 2.\r\n" );
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document.write( "That's the answer you were looking for.  But I'll continue:\r\n" );
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document.write( "To find that differential equation, we differentiate\r\n" );
document.write( "twice to get rid of the arbitrary constants:\r\n" );
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document.write( "y² = c1x + c2\r\n" );
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document.write( "2y*y' = c1\r\n" );
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document.write( " y*y' = c   where c = c1/2 \r\n" );
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document.write( "y*y\" + y'*y' = 0\r\n" );
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document.write( "y*y\" + (y')² = 0\r\n" );
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document.write( "That would be the 2nd order differential equation whose solution \r\n" );
document.write( "is all parabolas with their axis of symmetry being the x-axis.\r\n" );
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document.write( "Edwin
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