91Ó°ÊÓ

First of all, take the given function as,$\mathrm{Ï•}\left(\mathrm{x}\right)=\mathrm{y}$

Differentiating $\mathrm{Ï•}\left(\mathrm{x}\right)=\mathrm{y}={\mathrm{Ce}}^{3\mathrm{x}}+1$, with respect to x,$\frac{\mathrm{dy}}{\mathrm{dx}}=3{\mathrm{Ce}}^{3\mathrm{x}}$

$$\begin{gathered} \frac{\mathrm{dy}}{\mathrm{dx}}=3{\mathrm{Ce}}^{3\mathrm{x}}+3-3 \\ \frac{\mathrm{dy}}{\mathrm{dx}}=3\left({\mathrm{Ce}}^{3\mathrm{x}}+1\right)-3 \\ \frac{\mathrm{dy}}{\mathrm{dx}}=3\mathrm{y}-3 \\ \frac{\mathrm{dy}}{\mathrm{dx}}-3\mathrm{y}=-3 \end{gathered}$$

Which is identical to the given differential equation.

Hence,$\mathrm{Ï•}\left(\mathrm{x}\right)={\mathrm{Ce}}^{3\mathrm{x}}+1$ is a one-parameter family of solutions to $\frac{\mathrm{dy}}{\mathrm{dx}}-3\mathrm{y}=-3$, for any choice of the constant C.

When $c=0$

$y=1$(Represented with red colour)

When $c=1$

$y=e^{3x}+1$(Represented with blue colour)

When$c=-1$

$y=-e^{3x}+1$ (Represented with a blue-coloured dotted line)

When $c=2$

$y=2e^{3x}+1$(Represented with orange colour)

When$c=-2$

$y=-2e^{3x}+1$ (Represented with orange-coloured dotted line)

Graph representing the solution curves corresponding to$c=0,Â\pm 1,Â\pm 2$

Hence,$\mathit{Ï•}\left(x\right)\mathbf{=}\mathit{C}{\mathit{e}}^{\mathbf{3}\mathbf{x}}\mathbf{+}\mathbf{1}$ is a one-parameter family of solutions to$\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{-}\mathbf{3}\mathit{y}\mathbf{=}\mathbf{-}\mathbf{3}$, for any choice of the constant C.