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Образовательный студенческий форум > Высшая математика > Дифференциальные уравнения
jakata
Доброе время суток))
вот какая проблема:
Ускорение роста банковского вклада определяется из равенства a=t^2+t.
Найти закон изменения величины вклада, если в момент времени t=1 скорость его роста составляла V=5, а закон изменения выражался формулой S=1+5.
Подскажите пожалуйста с чего начать..
взять У-за величину вклада?
barklay
a= S''.

Следовательно, ДУ
S'' = t^2+t.
V(t) = S' = интеграл{t^2+t} = t^3/3 + t^2/2+C
V(1)=5/6+C = 5
Т.е C = 25/6
Итак
S' = t^3/3 + t^2/2+25/6
S = интеграл{t^3/3 + t^2/2+25/6} = t^4/12 + t^3/6+25t/6+C1.

Т.е.
S(t) = t^4/12 + t^3/6+25t/6+C1,
а уравнение касательной в точке t=1 есть Sk(t) = 1+5*t.
S(1) = Sk(1) = 6
53/12+C1 = 6
C1=19/12.

Откуда
S(t) = t^4/12 + t^3/6+25t/6+19/12.
wils
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