Question

Solve the differential equation $\frac{ds}{dt}=-s+t$

• A. $s=t+1+c{e}^{-t}$
• B. $s=t-1-c{e}^{-t}$
• C. $s=t-1+c{e}^{-t}$
• D. None of these.

Answer 1

Answer: (C)

$s=t-1+c{e}^{-t}$

$\frac{ds}{dt}=-s+t\Rightarrow \frac{ds}{dt}+s=t$
Here $P=1\Rightarrow \int Pdt=\int 1dt=t$ ...(1)
$\therefore IP=e^t$
Multiplying both sides of (1) by $I.F.$
$e^t\frac{ds}{dt}+s{e}^t=t{e}^t$
$\Rightarrow \frac{d}{dt}\left(s{e}^t\right)=t{e}^t$
Integrating both sides
$s{e}^t=\int t{e}^{t}dt+c=(t-1)e^t+c$
$\Rightarrow s=t-1+c{e}^{-t}$