Program Description: Purpose of problem is to obtain the particular functionfor the differential equation ( x 2 − 1 ) y ″ + 2 x y ′ + 2 y = x 2 − 1 if the complementary function is y c ( x ) = c 1 x + c 2 ( 1 + x 2 ) . Summary introduction: Problem will use a variation of the parameter method for the non-homogeneous equation y ″ + P ( x ) y ′ + Q ( x ) y = f ( x ) which has a complementary function y c ( x ) = c 1 y 1 ( x ) + c 2 y 2 ( x ) . The Wronskian of two independent solutions y 1 and y 2 of the associated homogeneous equation is W = W ( y 1 , y 2 ) and a particular solution of the non-homogeneous equation is y p ( x ) = − y 1 ( x ) ∫ y 2 ( x ) f ( x ) W ( x ) d x + y 2 ( x ) ∫ y 1 ( x ) f ( x ) W ( x ) d x .

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Chapter 3.5, Problem 62P

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Program Description: Purpose of problem is to obtain the particular functionfor the differential equation (x2−1)y″+2xy′+2y=x2−1 if the complementary function is yc(x)=c1x+c2(1+x2) .

Summary introduction: Problem will use a variation of the parameter method for the non-homogeneous equation y″+P(x)y′+Q(x)y=f(x) which has a complementary function yc(x)=c1y1(x)+c2y2(x) . The Wronskian of two independent solutions y1 and y2 of the associated homogeneous equation is W=W(y1,y2) and a particular solution of the non-homogeneous equation is yp(x)=−y1(x)∫ y 2( x)f( x)W( x)dx+y2(x)∫ y 1( x)f( x)W( x)dx .

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