document.write( "Question 844495: Solve the general solution of y'''+ 4y'= x+3cosx by undertermine
\n" ); document.write( "coeeficient.\r
\n" ); document.write( "\n" ); document.write( "Find the particular solution of y'''+y'=tanx -pi/2\n" ); document.write( " by variation \n" ); document.write( "
Algebra.Com's Answer #508900 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "First we find the complementary solution of the \r\n" );
document.write( "complementary homogeneous differential equation:\r\n" );
document.write( "\r\n" );
document.write( "y''' + 4y' = 0\r\n" );
document.write( "\r\n" );
document.write( "The auriliary polynomial equation is\r\n" );
document.write( "\r\n" );
document.write( "r³ + 4r = 0\r\n" );
document.write( "r(r²+4) = 0\r\n" );
document.write( "r=0, r=±2i, which we think of as \"0%B12i\"\r\n" );
document.write( "\r\n" );
document.write( "\"y%5Bc%5D\"\"%22%22=%22%22\"\"c%5B1%5De%5E%280x%29\"\"%22%22%2B%22%22\"\"e%5E%280x%29%28c%5B2%5Dcos%282x%29%2Bc%5B3%5Dsin%28x%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"y%5Bc%5D\"\"%22%22=%22%22\"\"c%5B1%5D%2A1\"\"%22%22%2B%22%22\"\"1%2A%28c%5B2%5Dcos%282x%29%2Bc%5B3%5Dsin%28x%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"y%5Bc%5D\"\"%22%22=%22%22\"\"c%5B1%5D%2Bc%5B2%5Dcos%282x%29%2Bc%5B3%5Dsin%28x%29\"\r\n" );
document.write( "\r\n" );
document.write( "Now we look for a particular solution to the original differential equation.\r\n" );
document.write( "\r\n" );
document.write( "The right side is x + 3cos(x)\r\n" );
document.write( "\r\n" );
document.write( "That would ordinarily make us think of this particular solution;\r\n" );
document.write( "\r\n" );
document.write( "Ax + Bcos(x) + Csin(x)\r\n" );
document.write( "\r\n" );
document.write( "We do not need a constant term because the complementary solution\r\n" );
document.write( "already has c1.\r\n" );
document.write( "\r\n" );
document.write( "However there is a \"conflict\" with the term x in the right side\r\n" );
document.write( "and the term Ax in that choice for a particular solution. So we\r\n" );
document.write( "must change that term by multiplying the term Ax by x, getting Ax².  \r\n" );
document.write( "That means we must still have a term in x.  So we redo the assumed \r\n" );
document.write( "particular solution to this:\r\n" );
document.write( "\r\n" );
document.write( "yp = Ax² + Bx + Ccos(x) + Dsin(x)\r\n" );
document.write( "\r\n" );
document.write( "yp' = 2Ax + B - Csin(x) + Dcos(x) [we can ignore the B, since we have c1]\r\n" );
document.write( "\r\n" );
document.write( "yp\" = 2A - Ccos(x) - Dsin(x)   [we can ignore the 2A, since we have c1]\r\n" );
document.write( "\r\n" );
document.write( "yp''' =  Csin(x) - Dcos(x)\r\n" );
document.write( "\r\n" );
document.write( "No we'll line up the terms of the original differential equation:\r\n" );
document.write( "\r\n" );
document.write( " yp''' =        Csin(x) -  Dcos(x)\r\n" );
document.write( "\r\n" );
document.write( "+4yp'  = 8Ax - 4Csin(x) + 4Dcos(x)\r\n" );
document.write( "--------------------------------------------------\r\n" );
document.write( "x + 3cos(x)       = 8Ax - 3Csin(x) + 3Dcos(x)\r\n" );
document.write( "\r\n" );
document.write( "Equating coefficients of x:  1 = 8A, so A = \"1%2F8\"\r\n" );
document.write( "\r\n" );
document.write( "Equating constants: 0 = 4B, so B=0\r\n" );
document.write( "\r\n" );
document.write( "Equating coefficients of cos(x): 3 = 3D, so D = 1  \r\n" );
document.write( "\r\n" );
document.write( "Equating coefficients of sin(x): 0 = -3C, so C = 0\r\n" );
document.write( "\r\n" );
document.write( "Particular solution:\r\n" );
document.write( "\r\n" );
document.write( "yp = Ax² + Bx + Ccos(x) + Dsin(x)\r\n" );
document.write( "\r\n" );
document.write( "yp = \"expr%281%2F8%29x%5E2+%2B+0x+%2B+0cos%28x%29+%2B+1sin%28x%29\"\r\n" );
document.write( "\r\n" );
document.write( "yp = \"expr%281%2F8%29x%5E2+%2B+sin%28x%29\"\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "General solution: \r\n" );
document.write( "\r\n" );
document.write( "\"y\"\"%22%22=%22%22\"\"y%5Bc%5D\"\"%22%22%2B%22%22\"\"y%5Bp%5D\"\r\n" );
document.write( "\r\n" );
document.write( "\"y\"\"%22%22=%22%22\"\"c%5B1%5D%2Bc%5B2%5Dcos%282x%29%2Bc%5B3%5Dsin%28x%29\"\"%22%22%2B%22%22\"\"expr%281%2F8%29x%5E2+%2B+sin%28x%29\".\r\n" );
document.write( "\r\n" );
document.write( "--------------------------------------\r\n" );
document.write( "\r\n" );
document.write( "If I have time I'll do the other one by variation of parameters later.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );