NO(x2y+y2)dx+(y3−x3)dy=0xy2+y41((x2y+y2)dx+(y3−x3)dy)=0xy2+y4x2y+y2dx+xy2+y4y3−x3dy=0∂y∂(xy2+y4x2y+y2)=∂y∂(xy+y3x2+y)=(xy+y3)2(xy+y3)(1)−(x2+y)(x+3y2)=(xy+y3)2xy+y3−x3−xy−3x2y2−3y3=(xy+y3)2xy−2y3−x3−xy−3x2y2∂x∂(xy2+y4y3−x3)=∂x∂(xy2+y4y3−x3)=(xy+y3)2(xy2+y4)(−3x2)−(y3−x3)(y2)=(xy+y3)2y(−3x3y−3x2y3−y4+x3y)=(xy+y3)2−2x3y2−3x2y4−y5Since∂x∂(xy2+y4y3−x3)=∂y∂(xy2+y4x2y+y2)∴The integrating factor is notxy2+y41
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