Question

In Problems $\mathbf{11}\mathbf{-}\mathbf{14}$verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution.

$\mathit{y}\mathbf{"}\mathbf{+}\mathit{y}\mathbf{=}\mathit{t}\mathit{a}\mathit{n}\mathbf{;}\mathbf{=}\mathbf{-}\left(cosx\right)\mathbf{}\mathit{I}\mathit{n}\left(secx+tanx\right)$.

Short answer

Answer:

The indicated function is an explicit solution of the given differential equation.

Step-by-step solution

Define an explicit function.

In explicit solution unknown functions (dependable variables) is indicated (expressed) in terms of undependable variable and constants. This is known as “explicit solution”.

Determine the derivatives of the function.

Let the given function be $y"+y=\tan x$.

Then, the first derivative of the function is,

$\mathit{y}\mathbf{\text{'}}\mathbf{=}\mathbf{-}\left(cosx\right)\mathit{s}\mathit{e}\mathit{c}\mathbf{}\mathit{x}\mathbf{}\mathbf{+}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{x}\mathbf{}\mathit{I}\mathit{n}\left(secx+tanx\right)$

Using the product rule, the second derivative of the function is,

$$\begin{gathered} \mathit{y}\mathbf{"}\mathbf{=}\left[-cosx\left(secxtanx\right)+sinxsecx\right]\mathbf{+}\left[sinxsecx+cosxIn\left(secx+tanx\right)\right] \\ \mathit{y}\mathbf{"}\mathbf{=}\mathbf{-}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{x}\mathbf{}\left(secxtanx\right)\mathbf{+}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{x}\mathbf{}\mathit{I}\mathit{n}\left(secx+tanx\right)\mathbf{+}\mathbf{2}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{x}\mathbf{}\mathit{s}\mathit{e}\mathit{c}\mathbf{}\mathit{x} \end{gathered}$$

Determine the explicit solution and its interval.

Substitute y, y' and y" in the left-hand side of the differential equation.

$$\begin{gathered} -\cos x\left(secx\tan x\right)+\cos xin\left(secx+\tan x\right)+2\sin xsecx+\left[-\left(\cos x\right)in\left(secx+\tan x\right)\right]=\tan \\ -\cos x\left(secx\tan x\right)+2\sin xsecx=\tan \\ -\frac{1}{secx}\left(secx\tan x\right)+2\sin x\frac{1}{\cos x}=\tan \\ -\tan x÷2\tan x=\tan \\ \tan x=\tan \end{gathered}$$

That is the same as the right-hand side of the differential equation for every real number $\mathbf{x}$. Thus, the indicated function is an explicit solution of the given differential equation.