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Консультация № 123134
Обсуждение
Неизвестный
14.02.2008, 17:16
общий
это ответ
Здравствуйте, Shariggg!
y‘=(x^2+xy-y^2)/(2x^2-2xy)
Однородное диф. ур-ние первого пор.
Замена y=tx
y‘=t‘x+t
t‘x+t=(x^2+t*x^2-t^2*x^2)/(2x^2-2tx^2)
t‘x+t=(1+t-t^2)/(2-2t)
-((2t-1)+1)dt/(t^2-t+1)=dx/x
-Int[d(t^2-t+1)/t^2-t+1]+Int[dt/(t^2-t+1)]=Int[dx/x]+C (сначала внесли под знак дифференциала 2t-1)
Выделим полный квадрат в знаменателе второго слагаемого:
-ln|t^2-t+1|+Int[d(t-1/2)/((t-1/2)^2+3/4)]=lncx
-ln|t^2-t+1|+2/sqrt3 *arctg[2/sqrt3*(t-1/2)]=lncx
Теперь осталось лишь подставить вместо t выражение y/x.
y‘=(x^2+xy-y^2)/(2x^2-2xy)
Однородное диф. ур-ние первого пор.
Замена y=tx
y‘=t‘x+t
t‘x+t=(x^2+t*x^2-t^2*x^2)/(2x^2-2tx^2)
t‘x+t=(1+t-t^2)/(2-2t)
-((2t-1)+1)dt/(t^2-t+1)=dx/x
-Int[d(t^2-t+1)/t^2-t+1]+Int[dt/(t^2-t+1)]=Int[dx/x]+C (сначала внесли под знак дифференциала 2t-1)
Выделим полный квадрат в знаменателе второго слагаемого:
-ln|t^2-t+1|+Int[d(t-1/2)/((t-1/2)^2+3/4)]=lncx
-ln|t^2-t+1|+2/sqrt3 *arctg[2/sqrt3*(t-1/2)]=lncx
Теперь осталось лишь подставить вместо t выражение y/x.
Форма ответа
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