document.write( "Question 964019: please any one solve this differential equation :
\n" ); document.write( "(2ax+x^2)dy/dx=a^2+2ax \n" ); document.write( "

Algebra.Com's Answer #588916 by Edwin McCravy(20054) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"%282ax%2Bx%5E2%29expr%28dy%2Fdx%29\"$ $\"%22%22=%22%22\"$ $\"a%5E2%2B2ax\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"dy\"$ $\"%22%22=%22%22\"$ $\"expr%28%28a%5E2%2B2ax%29%2F%282ax%2Bx%5E2%29%29dx\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"int%28dy%29\"$ $\"%22%22=%22%22\"$ $\"int%28expr%28%28a%5E2%2B2ax%29%2F%282ax%2Bx%5E2%29%29dx%29\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"y\"$ $\"%22%22=%22%22\"$ $\"int%28expr%28%28a%28a%2B2x%29%29%2F%282ax%2Bx%5E2%29%29dx%29\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"y\"$ $\"%22%22=%22%22\"$ $\"a%2Aint%28expr%28%28a%2B2x%29%2F%282ax%2Bx%5E2%29%29dx%29\"$ \r
\n" ); document.write( "\n" ); document.write( "If the numerator were the derivative of the denominator, we
\n" ); document.write( "would be able to integrate using $\"int%28du%2Fu%29=ln%28abs%28u%29%29%2BC\"$ \r
\n" ); document.write( "\n" ); document.write( "The derivative of the denominator is 2a+2x.
\n" ); document.write( "The numerator a+2x only needs to have \"a\" added to it to become that
\n" ); document.write( "So in the numerator we will add \"a\" and subtract \"a\"\r
\n" ); document.write( "\n" ); document.write( " $\"y\"$ $\"%22%22=%22%22\"$ $\"a%2Aint%28expr%28%28red%28a%29%2Ba%2B2x-red%28a%29%29%2F%282ax%2Bx%5E2%29%29dx%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"y\"$ $\"%22%22=%22%22\"$ $\"a%2Aint%28expr%28%282a%2B2x-red%28a%29%29%2F%282ax%2Bx%5E2%29%29dx%29\"$ \r
\n" ); document.write( "\n" ); document.write( "Write as the difference of two fractions and integrate each:\r
\n" ); document.write( "\n" ); document.write( " $\"y\"$ $\"%22%22=%22%22\"$

\r
\n" ); document.write( "\n" ); document.write( " $\"y\"$ $\"%22%22=%22%22\"$

\r
\n" ); document.write( "\n" ); document.write( " $\"y\"$ $\"%22%22=%22%22\"$ $\"a%2Aint%28expr%28%282a%2B2x%29%2F%282ax%2Bx%5E2%29%29dx%29\"$ $\"%22%22-%22%22\"$ $\"a%2Aint%28expr%28a%2F%282ax%2Bx%5E2%29%29dx%29\"$ \r
\n" ); document.write( "\n" ); document.write( "Use the natural log formula on the first integral, and take the
\n" ); document.write( "constant \"a\" outside the second integral:\r
\n" ); document.write( "\n" ); document.write( " $\"y\"$ $\"%22%22=%22%22\"$ $\"a%2Aln%28abs%282ax%2Bx%5E2%29%29\"$ $\"%22%22-%22%22\"$ $\"a%5E2%2Aint%28expr%281%2F%282ax%2Bx%5E2%29%29dx%29\"$ \r
\n" ); document.write( "\n" ); document.write( "To the side, let's complete the square in the denominator by
\n" ); document.write( "adding and subtrracting a²:\r
\n" ); document.write( "\n" ); document.write( " $\"2ax%2Bx%5E2\"$ $\"%22%22=%22%22\"$ $\"x%5E2%2B2ax%2Ba%5E2-a%5E2\"$ $\"%22%22=%22%22\"$ $\"%28x%2Ba%29%5E2-a%5E2\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"y\"$ $\"%22%22=%22%22\"$ $\"a%2Aln%28abs%282ax%2Bx%5E2%29%29\"$ $\"%22%22-%22%22\"$ $\"a%5E2%2Aint%28expr%281%2F%28%28x%2Ba%29%5E2-a%5E2%29%29dx%29\"$ \r
\n" ); document.write( "\n" ); document.write( "To do the remaining integral we use the formula $\"int%28+du%2F%28u%5E2-a%5E2%29+%29\"$ $\"%22%22=%22%22\"$ $\"expr%28+1%2F%282a%29+%29ln%28abs%28+%28u-a%29%2F%28u%2Ba%29%29+%29%2BC\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"y\"$ $\"%22%22=%22%22\"$ $\"a%2Aln%28abs%282ax%2Bx%5E2%29%29\"$ $\"%22%22-%22%22\"$

\r
\n" ); document.write( "\n" ); document.write( " $\"y\"$ $\"%22%22=%22%22\"$ $\"a%2Aln%28abs%282ax%2Bx%5E2%29%29\"$ $\"%22%22-%22%22\"$ $\"expr%28+a%2F2+%29ln%28abs%28+x%2F%28x%2B2a%29%29+%29%2BC\"$ \r
\n" ); document.write( "\n" ); document.write( "Edwin \n" ); document.write( "

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