Question

Consider the differential equation$\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{=}\mathbf{5}\mathbf{-}\mathit{y}$.

(a) Either by inspection or by the method suggested in Problems 37–40, and a constant solution of the DE.

(b) Using only the differential equation, find intervals on the y-axis on which a non constant solution$\mathit{y}\mathbf{=}\mathit{\varphi }\mathbf{\left(}\mathit{x}\mathbf{\right)}$ is increasing. Find intervals on the y-axis on which$\mathit{y}\mathbf{=}\mathit{\varphi }\mathbf{\left(}\mathit{x}\mathbf{\right)}$ is decreasing.

Short answer

a. The constant solution is $y=5$.

b. The solution is increasing for $y<5$and decreasing for $y>5$.

Step-by-step solution

Definition of differential equation.

A differential equation is defined as the derivative function of one or more unknown functions (dependable variables) with respect to one or more undependable variables.

Solution for part a)

The function is constant for $\frac{dy}{dx}=0$ , hence, we have:

$\frac{dy}{dx}=5-y$

So,

$$\begin{gathered} 5-y=0 \\ y=5 \end{gathered}$$

Therefore, the constant solution is $y=5$, this is easy to verify since for $y=5$:

$$\begin{gathered} 0=\frac{dy}{dx} \\ 5-5=0 \end{gathered}$$

Solution for part b)

Let $f$be a function and let $D$be its domain. If $f\text{'}\left(x\right)>0$for all values $x\in D$, then $f$is an increasing function.

Therefore, the solution $y=\varphi \left(x\right)$is increasing when $\frac{dy}{dx}>0$, we have:

$\frac{dy}{dx}=5-y>0$

So, y<5

Similarly the solution is decreasing when $\frac{dy}{dx}<0$, hence

$\frac{dy}{dx}=5-y<0$

So, y>0.

Hence, the constant solution is y=5, the increasing when y>5 and decreasing when y<5.