Mathematics - Differential Equation Question with Solution | TestHub

$\frac{dp\left(t\right)}{dt}=\frac{1}{2}\{p\left(t\right)-400\}$

$p\left(0\right)=100$

$\frac{2dp\left(t\right)}{p\left(t\right)-400}=dt$

Integrating,

$\Rightarrow {\int }_0^{p\left(t\right)}\frac{\mathit{dp}\left(t\right)}{p\left(t\right)-400}={\int }_0^t\frac{1}{2}\mathit{dt}\Rightarrow {\left(\ln \left|p\left(t\right)-400\right|\right)}_0^{p\left(t\right)}=\frac{t}{2}$

$\Rightarrow \ln \left|\frac{p\left(t\right)-400}{300}\right|=\frac{t}{2}$

$\Rightarrow \left|p\left(t\right)-400\right|=300e^{\frac{t}{2}}$

$\Rightarrow 400-p\left(t\right)=300e^{\frac{t}{2}}\mathit{} (\therefore p\left(t\right)<400)$

$\Rightarrow p\left(t\right)=400-300e^{\frac{t}{2}}$