Question

What are the intervals of definition of the general solutions in Problems 1,7,15, and 18? Discuss why the interval of definition of the general solution in Problem 28 is not $\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{\infty }\mathbf{)}$.

Short answer

The intervals of definition of the general solutions by given equation is:

Problem 1:$\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$

Problem 7:$\left(-\infty ,\infty \right)$

Problem 15:$\left(0,\infty \right)$

Problem 18: $\left(-1,1\right)$

Problem 24: $-\frac{\pi }{2}<\text{log}x<\frac{{\displaystyle \pi }}{{\displaystyle 2}}{e}^{-\pi /2}<x<e^{\pi /2}\text{.}$

Step-by-step solution

Deriving the given equation by using the intervals

The problem 1 is given as$y\text{'}\text{'}+y=\text{sec}x$.

The interval of definition of the general solutions in Problems 1 should be$\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$

The problem 7 is given as

The interval of definition of the general solutions in Problems 7 should be $\left(-\infty ,\infty \right)$.

The problem 9 is given as:

$y\text{'}\text{'}-4y=\frac{e^{2x}}{x}$

The interval of definition of the general solutions in Problems 9 should be $\left(0,\infty \right)$

The problem 18 is given as:

$4y\text{'}\text{'}-4y\text{'}+y=e^{x/2}\sqrt{1-x^2}$

The interval of definition of the general solutions in Problems 18 should be $\left(-1,1\right)$.

Find the interval by using differential equation

The problem 24 with linearly independent solutions is given as:

$$\begin{gathered} x^{2}y\text{'}\text{'}+xy\text{'}+y=\text{sec}\left(\text{log}x\right) \\ {y}_1=\text{cos}\left(\text{log}x\right) \\ {y}_2=\text{sin}\left(\text{log}x\right) \end{gathered}$$

The general solution is given as:

$y=c_1\text{cos}\left(\text{log}x\right)+c_2\text{sin}\left(\text{log}x\right)+\text{cos}\left(\text{log}x\right)\text{log}\left|\text{cos}\left(\text{log}x\right)\right|+\text{log}\left(x\right)\text{sin}\left(\text{log}x\right)$

For the interval $-\frac{\pi }{2}<\text{log}x<\frac{{\displaystyle \pi }}{{\displaystyle 2}}or{e}^{-\pi /2}<x<e^{\pi /2}$

The function log x has the interval $-\frac{\pi }{2}<\text{log}x<\frac{\pi }{2}$because of the sec function in the differential equation.