Question

In Problems 35–38 the graph of a member of a family of solutions of a second-order differential equation ${\mathit{d}}^{\mathbf{2}}\mathit{y}\mathbf{/}\mathit{d}{\mathit{x}}^{\mathbf{2}}\mathbf{=}\mathit{f}\mathbf{(}\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{y}\mathbf{\text{'}}\mathbf{)}$is given. Match the solution curve with at least one pair of the following initial conditions.

$\begin{array}{l}\mathbf{a}\mathbf{)}\mathbf{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{2}\\ \mathbf{b}\mathbf{\left)}\mathbf{y}\mathbf{\right(}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{=}\mathbf{-}\mathbf{4}\\ \mathbf{c}\mathbf{)}\mathbf{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{2}\\ \mathbf{d}\mathbf{)}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{2}\\ \mathbf{e}\mathbf{)}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\\ \mathbf{f}\mathbf{)}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{4}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{2}\end{array}$

FIGURE 1.2.9 Graph for Problem 37

Short answer

The solutions (c) $\mathrm{y}\left(1\right)=1,\mathrm{y}\text{'}\left(1\right)=2$and (d)$\mathrm{y}\left(0\right)=-1,\mathrm{y}\text{'}\left(0\right)=2$ match the given curve.

Step-by-step solution

Definition of differential equation

Differential Equation is defined as the equation that consists of the derivatives of one or more dependent functions in respect to one or more independent functions.

Check the slope of the function

We can see that the graph passes through $(-1,-2),(0,-1)\text{\hspace{0.17em}and\hspace{0.17em}}(1,1)$which corresponds to the following conditions.

$\begin{array}{l}y(-1)=-2\\ y\left(0\right)=-1\\ y\left(1\right)=1\end{array}$

It seems that the condition matches (a), (c) (d) or (e) since $y\left(0\right)=-1$ and$y\left(1\right)=1$ .

From the graph, role="math" localid="1663825751562" $\mathrm{y}\text{'}\left(0\right)$is positive and $\mathrm{y}\text{'}\left(1\right)$is positive.

We check the slope of the function at x=0, which is positive and the slope of the function at x=1, which is positive.

From (c), $y\text{'}\left(1\right)=2$and From (d),$y\text{'}\left(0\right)=2$

So the information from the graph eliminates (a) and (e). Since the slope at x=0 and x=1 are about the same, (c) and (d) are both acceptable answers.

Therefore, the solutions (c) $\mathrm{y}\left(1\right)=1,\mathrm{y}\text{'}\left(1\right)=2$and (d) $\mathrm{y}\left(0\right)=-1,\mathrm{y}\text{'}\left(0\right)=2$match the given curve.