Question

Find a differential equation of the form$\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{k}\mathit{x}$for which $\mathit{x}\left(t\right)\mathbf{=}{\mathbf{3}}^{\mathbf{t}}$is a solution.

Short answer

The differential equation is$\frac{dx}{dt}=\mathrm{In}\left(3\right)x$ .

Step-by-step solution

Definition of the differential equation

Consider the differential equation$\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathit{k}\mathit{x}$ with initial value ${\mathit{x}}_{\mathbf{0}}$(k is an arbitrary constant), then the solution is$\mathit{x}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}{\mathit{x}}_{\mathbf{0}}{\mathit{e}}^{\mathbf{k}\mathbf{t}}$ .

Find the differential equation

Since$x\left(t\right)=3^t$is a solution of the differential equation.

Therefore, the equation can be represented as follows:

role="math" localid="1659682128421" $$\begin{gathered} \frac{d}{dt}\left(3^t\right)=In\left(3\right)3^t,\mathrm{where}\mathrm{k}=\mathrm{In}\left(3\right) \\ \frac{d}{dt}\left(3^t\right)=In\left(3\right)x\left(t\right)\left\{\because x\left(t\right)=3^t\right\} \end{gathered}$$

Hence, the differential equation is given as follows.

$\frac{dx}{dt}=\mathrm{In}\left(3\right)x$

Thus, the differential equation is $\frac{dx}{dt}=\mathrm{In}\left(3\right)x$.