Question

In Problems 1–22 Solve the given differential equation by separationof variables.

$\frac{\mathbf{\text{dQ}}}{\mathbf{\text{dt}}}\mathbf{\text{=k(Q-70)}}$

Short answer

$Q=C{e}^{kt}+70$

Step-by-step solution

Step 1:Definition of separable equation

A first-order differential equation of the form$\frac{dy}{dx}=g\left(x\right)h\left(y\right)$is said to be separable or to have separable variables.

Step 2:Separate the variables and integration

Separate the variables

$\frac{dQ}{Q-70}=kdt$

Integrate both sides

$\begin{array}{l}\int \frac{dQ}{Q-70}=\int kdt\\ \ln |Q-70|=kt+C_1\end{array}$

Take to both sides

$e^{\ln |Q-70|}=e^{kt+C_1}$

Remember that$e^{a+b}=e^a{e}^b$

So

$e^{\ln |Q-70|}=e^{kt}{e}^{C_1}$

Where $e^{\ln |Q-70|}=Q-70$and$e^{C_1}=C$

So

$\begin{array}{c}Q-70=C{e}^{kt}\\ Q=C{e}^{kt}+70\end{array}$

Therefore the solution is${\mathbf{\text{Q=Ce}}}^{\mathbf{\text{kt}}}\mathbf{\text{+70}}$