document.write( "Question 1188305: Find the family of solutions of the following equation using integrating factors
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Algebra.Com's Answer #819367 by Edwin McCravy(20064) $\"\"$ $\"About$

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document.write( "The trick here is to get the left side into the expression of either the product expression of \r\n" );
document.write( "the differential of uv, which is:\r\n" );
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document.write( "or the quotient expression for the differential of , which is:\r\n" );
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document.write( "We notice by inspection that the third term has x³ and the first term has 3x²dx, which is the \r\n" );
document.write( "differential of x³.  So we will get those two terms together and the other term\r\n" );
document.write( "on the other side:\r\n" );
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document.write( "The product rule isn't going to work because there's a MINUS sign between the terms, not a \r\n" );
document.write( "PLUS sign.  So, we think of trying to make the left side into the form \r\n" );
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document.write( ", the differential form for \r\n" );
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document.write( "The du can be the 3x²dx (with u as x³), and the v can be the y. So we'll write the y first \r\n" );
document.write( "in the first term:  \r\n" );
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document.write( "Now on the left we have the numerator of the differential form for .\r\n" );
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document.write( "Now all we need to make the left side into the quotient differential form of  is to \r\n" );
document.write( "divide both sides of the equation through by y²:\r\n" );
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document.write( "So the integrating factor used here is .\r\n" );
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document.write( "Now we can integrate both sides of the equation:\r\n" );
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document.write( "The whole left side integrates all together as the quotient , and, the right side \r\n" );
document.write( "integrates as  and we must add an arbitrary constant:\r\n" );
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document.write( "Multiply through by 4y\r\n" );
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document.write( "[We just write C because 4 times an arbitrary constant is just another arbitrary\r\n" );
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document.write( "Edwin

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