Question

In Problems 45–48 without solving, match a solution curve of $\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathit{y}\mathbf{=}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}$shown in the figure with one of the following functions:

$$\begin{gathered} \mathbf{\left(}\mathit{i}\mathbf{\right)}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,} \\ \mathbf{\left(}\mathit{i}\mathit{i}\mathbf{\right)}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{e}}^{\mathbf{-}\mathbf{x}}\mathbf{,} \\ \mathbf{\left(}\mathit{i}\mathit{i}\mathit{i}\mathbf{\right)}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{e}}^{\mathbf{x}}\mathbf{,} \\ \mathbf{\left(}\mathit{i}\mathit{v}\mathbf{\right)}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathbf{2}\mathit{x}\mathbf{,} \\ \mathbf{\left(}\mathit{v}\mathbf{\right)}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{e}}^{\mathbf{x}}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{x}\mathbf{,} \\ \mathbf{\left(}\mathit{v}\mathit{i}\mathbf{\right)}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{x}\mathbf{,} \end{gathered}$$

Short answer

$\left(v\right)f\left(x\right)=e^{x}sinx$

Step-by-step solution

Match the solution

The graph for problem 45 shown in the figure 4.4 .1 in the book is being matched by the differential equation shown in (v) which is $y\text{'}\text{'}+y=e^{x}sinx$ which has the solution $y\left(x\right)=Asinx+Bcosx+C{e}^{x}sinx+D{e}^{x}cosx$where the function $e^x$ increases the value of rapidly for the positive values of x.

$\left(v\right)f\left(x\right)=e^{x}sinx$

Conclusion

The solution is $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{e}}^{\mathbf{x}}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{x}$