Q17 E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Show that $\mathbf{\varphi }\left(x\right)\mathbf{=}{\mathbf{Ce}}^{\mathbf{3}\mathbf{x}}\mathbf{+}\mathbf{1}$ is a solution to $\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{-}\mathbf{3}\mathbf{y}\mathbf{=}\mathbf{-}\mathbf{3}$ for any choice of the constant C. Thus, ${\mathbf{Ce}}^{\mathbf{3}\mathbf{x}}\mathbf{+}\mathbf{1}$ is a one-parameter family of solutions to the differential equation. Graph several of the solution curves using the same coordinate axes.

Step-by-Step Solution

Answer

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

The graph for the solution curves is drawn.

Step 1: Taking the given function as y

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

Step 2: Differentiating the given function with respect to x

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

Step 3: Simplification of the differential equation obtained in step 2

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

Which is identical to the given differential equation.

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

Step 4: Representing the solution curves on a graph

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 $c=0,\pm 1,\pm 2$

Hence, $\mathit{\varphi }\left(x\right)\mathbf{=}\mathit{C}{\mathit{e}}^{\mathbf{3}\mathbf{x}}\mathbf{+}\mathbf{1}$ 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.