Solve differential equation dxdy+xy=11+y2
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dxdy+p(y)x=q(y)Compare the equation dxdy+p(y)x=q(y) to dxdy+xy=11+y2 and obtain p(y)=1y, q(y)=11+y2I.F.=e∫p(y)dy=elny=y xe∫p(y)dy=∫e∫p(y)dyq(y)dy+Cwhere C is arbitrary constant of equationxe∫p(y)dy=∫e∫p(y)dyq(y)dy+Cxy=∫y11+y2dy+Cxy=12∫2y1+y2dy+Cxy=12∫(1+y2)′1+y2dy+C [∵(1+y2)′=2y]xy=12(21+y2)+C [∵∫f(y)f(y)dy=2f(y)]xy=1+y2+Cx=1+y2+Cyx=1+y2y+Cy
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Convert the differential equation u″−u′−2u=e−5t into a system of first order equations by letting x=u, y=u'x′=?y′=?
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