16. dQ/dt=k(Q-70) solve by separable method

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16. dQ/dt=k(Q-70) solve by separable method.

To solve the differential equation \frac{dQ}{dt}=k(Q-70) by separable method, we need to separate the variables Q and t on opposite sides of the equation and integrate both sides.

Starting with the given differential equation:

    \begin{equation*} \frac{dQ}{dt} = k(Q-70) \end{equation*}

We can divide both sides by (Q-70) to obtain:

    \begin{equation*} \frac{1}{Q-70}dQ = kdt \end{equation*}

Now we can integrate both sides with respect to their respective variables:

    \begin{align*} \int \frac{1}{Q-70}dQ &= \int kdt \end{align*}

    \begin{align*}\ln|Q-70| &= kt + C_1 \end{align*}

    \begin{align*} Q-70 &= e^{kt+C_1} \end{align*}

    \begin{align*}Q &= Ce^{kt} + 70 \end{align*}

where C is a constant determined by the value of the constant of integration C_1.

Therefore, the general solution to the given differential equation is:

    \begin{equation*} Q = Ce^{kt} + 70 \end{equation*}

where C is a constant determined by the initial conditions.

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