Question

Use the concept that y = c, −[infinity] < x < [infinity], is a constant function if and only if y' = 0 to determine whether the given differential equation possesses constant solutions. (Enter all constant solutions below. If there are no constant solutions, enter NONE.)7xy' + 5y = 15y = ___.

Answer 1

Answer:

$y = 3$

Step-by-step explanation:

By using this concept, we can determine whether the given differential equation has constant solutions.  Consider the differential equation

                                           $7xy'+5y = 0$

Let

                                $y(x) = c, \; -\infty < x < \infty$

be a constant function, where $c$ is a constant. Then,

                                               $y' = 0$

Substituting $0$ for $y'$ and $c$ for $y$ in the given equation gives

                          $7x \cdot 0 + 5 \cdot c = 15 \iff 5 c = 15 \iff c = 3$

Therefore,  straight line  $y = 3$  is a constant solution of the given differential equation.