Solve the differential equation $\frac { d Q } { d t } = 200 - 0.2 Q$ subject to $Q ( 0 ) = 1500$ .
A) $Q ( t ) = 500 e ^ { - 0.2 t } + 1000$
B) $Q ( t ) = - 500 e ^ { - 0.2 t } + 1000$
C) $Q ( t ) = 1300 e ^ { - 0.2 t } + 200$
D) $Q ( t ) = - 1300 e ^ { - 0.2 t } + 200$

Question

Quizplus · Accepted Answer

This differential equation is a first-order linear ordinary differential equation. Solving it involves finding an integrating factor or using separation of variables. The general solution to such an equation is $Q(t) = Q_e + (Q_0 - Q_e)e^{-kt}$, where $Q_e$ is the equilibrium quantity (the value of $Q$ when $\frac{dQ}{dt} = 0$), $Q_0$ is the initial quantity, and $k$ is the rate constant. Here, $Q_e = 1000$ (since setting $\frac{dQ}{dt} = 0$ gives $200 - 0.2Q = 0$, solving for $Q$ gives $Q = 1000$), $k = 0.2$, and $Q_0 = 1500$. Plugging these values into the general solution formula gives $Q(t) = 1000 + (1500 - 1000)e^{-0.2t} = 500e^{-0.2t} + 1000$.

Solve the Differential Equation dQdt=200−0.2Q\frac { d Q } { d t } = 200 - 0.2 QdtdQ​=200−0.2Q

Solve the Differential Equation $\frac { d Q } { d t } = 200 - 0.2 Q$