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Консультация № 187871
07.06.2014, 08:18
78.80 руб.
07.06.2014, 21:42
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Обсуждение
07.06.2014, 22:53
общий
07.06.2014, 23:02
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07.06.2014, 23:31
общий
это ответ
Здравствуйте, Aleksandrkib!
xy'+y=xy2lnx.
Это уравнение Бернулли. Решим его методом вариации произвольной постоянной.
xy'+y=0,
dy/dx=-y/x,
dy/y=-dx/x.
Интегрируя, получим: lny=lnC-lnx [$8658$] y=C/x, С=const.
Общее решение исходного уравнения ищем в виде y=C(x)/x.
y'=C'(x)/x-C(x)/x2.
Подставляя у и у' в уравнение, имеем:
C'(x)-C(x)/x+C(x)/x=x(C(x)/x)2lnx,
dC(x)/dx=(C2(x)/x)lnx,
dC(x)/(C2(x))=lnxdx/x.
Интегрируя, получим: -1/C(x)=ln2x+C1 [$8658$] C(x)=-1/(ln2x+C1).
Итак, общее решение y=C(x)/x=-1/(xln2x+C1x).
xy'+y=xy2lnx.
Это уравнение Бернулли. Решим его методом вариации произвольной постоянной.
xy'+y=0,
dy/dx=-y/x,
dy/y=-dx/x.
Интегрируя, получим: lny=lnC-lnx [$8658$] y=C/x, С=const.
Общее решение исходного уравнения ищем в виде y=C(x)/x.
y'=C'(x)/x-C(x)/x2.
Подставляя у и у' в уравнение, имеем:
C'(x)-C(x)/x+C(x)/x=x(C(x)/x)2lnx,
dC(x)/dx=(C2(x)/x)lnx,
dC(x)/(C2(x))=lnxdx/x.
Интегрируя, получим: -1/C(x)=ln2x+C1 [$8658$] C(x)=-1/(ln2x+C1).
Итак, общее решение y=C(x)/x=-1/(xln2x+C1x).
5
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