Chapter 10, Problem 1RE

To determine

To fill: The given blank or answer true or false.

Answer to Problem 1RE

The given differential equation can be written as plane autonomous system. So, the statement is true.

Explanation of Solution

The given statement is “The second-order differential equation $x^{″}+f\left(x^{\prime }\right)+g\left(x\right)=0$ can be written as a plane autonomous system.”

Now, assume that $x^{\prime }=y$.                                                                                        (1)

So the given differential equation becomes

$\begin{array}{c}{x}^{″}+f\left(x^{\prime }\right)+g\left(x\right)=0\\ y^{\prime }+f\left(y\right)+g\left(x\right)=0\end{array}$

Which on further solving gives,

$y^{\prime }=-\left[f\left(y\right)+g\left(x\right)\right]$.                                                                                              (2)

So, the given differential equation can be written as

$x^{\prime }=y$ and $y^{\prime }=-\left[f\left(y\right)+g\left(x\right)\right]$

Thus, it can be written as a plane autonomous system.

Hence, the given differential equation can be written as plane autonomous system. So, the statement is true.