Question

The solution of the first order differential equation x' (t) = -3x(t), x(0) = $x_0$ is

• A. x(t) = $x_0{e}^{-3t}$
• B. x(t) = $x_0{e}^{-3}$
• C. x(t) = $x_0{e}^{-t/3}$
• D. x(t) = $x_0{e}^{-t}$

Answer 1

The correct option is A x(t) = $x_0{e}^{-3t}$

x'(t) = -3x(t)

${\int }_{x_0}^{x\left(t\right)}\frac{dx\left(t\right)}{x\left(t\right)}=-{\int }_0^{t}3.dt$

$\Rightarrow \left[lnx\right(t){]}_{x_0}^{x\left(t\right)}=-3(t-0)$

$\Rightarrow lnx\left(t\right)-ln{x}_0=-3t$

$\Rightarrow \frac{x\left(t\right)}{x_0}=e^{-3t}$

$\Rightarrow x\left(t\right)=x_0{e}^{-3t}$