Question

In Problems $\mathbf{15}\mathbf{-}\mathbf{18}$verify that the indicated function $\mathit{y}\mathbf{=}\mathit{\varphi }\left(x\right)$is an explicit solution of the given first-order differential equation. Proceed as in Example $\mathbf{6}$, by considering $\varphi$simply as a function and give its domain. Then by considering $\varphi$as a solution of the differential equation, give at least one interval $\mathbf{I}$of definition.

$\mathit{y}\mathbf{\text{'}}\mathbf{=}\mathbf{25}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{;}\mathit{y}\mathbf{=}\mathbf{5}\mathit{t}\mathit{a}\mathit{n}\mathbf{5}\mathit{x}$

Short answer

The indicated function is an explicit solution of the given differential equation and the interval is\(I:\left( { - \frac{\pi }{{10}},\frac{\pi }{{10}}} \right)\), and the domain is\(D:x \ne \frac{\pi }{{10}} + \frac{{n\pi }}{5}\).

Step-by-step solution

Define an explicit function.

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

Let the given function be $\mathit{y}\mathbf{=}\mathbf{5}\mathit{t}\mathit{a}\mathit{n}\mathbf{5}\mathit{x}$.

Then, the first derivative of the function is,

$y\text{'}=25se{c}^{2}5x$

Determine the explicit solution.

Substitute $y$and $y\text{'}$in the left-hand side of the differential equation.

$$\begin{gathered} \mathbf{25}\mathit{s}\mathit{e}{\mathit{c}}^{\mathbf{2}}\mathbf{5}\mathit{x}\mathbf{=}\mathbf{25}\mathbf{+}\mathbf{25}\mathit{t}\mathit{a}{\mathit{n}}^{\mathbf{2}}\mathbf{5}\mathit{x} \\ \mathbf{25}\mathit{s}\mathit{e}{\mathit{c}}^{\mathbf{2}}\mathbf{5}\mathit{x}\mathbf{=}\mathbf{25}\left(1+ta{n}^{2}5x\right) \\ \mathbf{25}\mathit{s}\mathit{e}{\mathit{c}}^{\mathbf{2}}\mathbf{5}\mathit{x}\mathbf{=}\mathbf{25}\mathit{s}\mathit{e}{\mathit{c}}^{\mathbf{2}}\mathbf{5}\mathit{x} \end{gathered}$$

That is same as the right-hand side of the differential equation. The indicated function is an explicit solution of the given differential equation.

Determine the domain and the interval.

Hence the domain of the solution while considering the solution as a function is,

$\left(Notethat\tan \theta isundefined,when\theta =\frac{\mathrm{\pi }}{2}\pm n\mathrm{\pi }\right)$

$$\begin{gathered} \mathrm{\pi x}/\frac{\mathrm{\pi }}{}+\mathrm{n} \\ \mathrm{x}\ne \frac{\mathrm{\pi }}{10}+\frac{\mathrm{n\pi }}{5} \\ \mathrm{D}:\mathrm{x}\ne \frac{\mathrm{\pi }}{10}+\frac{\mathrm{n\pi }}{5} \end{gathered}$$

And, then the interval is $I:\left(-\frac{\mathrm{\pi }}{10},\frac{\mathrm{\pi }}{10}\right)$.