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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 = ___.

Respuesta :

Answer:

[tex]y = 3[/tex]

Step-by-step explanation:

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

                                           [tex]7xy'+5y = 0[/tex]

Let

                                [tex]y(x) = c, \; -\infty < x < \infty[/tex]

be a constant function, where [tex]c[/tex] is a constant. Then,

                                               [tex]y' = 0[/tex]

Substituting [tex]0[/tex] for [tex]y'[/tex] and [tex]c[/tex] for [tex]y[/tex] in the given equation gives

                          [tex]7x \cdot 0 + 5 \cdot c = 15 \iff 5 c = 15 \iff c = 3[/tex]

Therefore,  straight line  [tex]y = 3[/tex]  is a constant solution of the given differential equation.

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