Solve the following differential equations : d d t =-k ( - _0 ) - Vidyadip

Question

Solve the following differential equations : $\frac{d \theta}{d t}=-k\left(\theta-\theta_0\right)$

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Answer

$
\begin{gathered}
\frac{d \theta}{d t}=-k\left(\theta-\theta_0\right) \\
\therefore \frac{1}{\theta-\theta_0} d \theta=-k d t
\end{gathered}
$
Integrating, we get
$
\begin{aligned}
& \int\frac{1}{\theta-\theta_0} d \theta=-k \int d t \\
& \therefore \log \left|\theta-\theta_0\right|=-k t+\log c_1 \\
& \therefore \log \left|\theta-\theta_0\right|-\log c_1=-k t \\
& \therefore \log \left|\frac{\theta-\theta_0}{c_1}\right|=-k t \\
& \therefore \frac{\theta-\theta_0}{c_1}=e^{-k t} \\
& \therefore \theta-\theta_0=c_1 e^{-k t} \\
& \therefore \theta-\theta_0=e^e \cdot e^{-k t}, \text { where } c_1=e^c \\
& \therefore \theta-\theta_0=e^{-k t+c}
\end{aligned}
$
This is the general solution.

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