Question

Consider the differential equation $\mathit{y}\mathbf{\text{'}}\mathbf{=}{\mathit{y}}^{\mathbf{2}}\mathbf{+}\mathbf{4}$

(a)Explain why there exist no constant solutions of the DE.

(b)Describe the graph of a solution$\mathit{y}\mathbf{=}\mathit{\varphi }\left(x\right)$. For example, can a solution curve have any relative extrema?

(c)Explain why$\mathit{y}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}$is the y-coordinate of a point of inflection of a solution curve.

(d)Sketch the graph of a solution$\mathit{y}\mathbf{=}\mathit{\varphi }\mathbf{\left(}\mathbf{x}\mathbf{\right)}$of the differential equation whose shape is suggested by parts (a) -(c).

Short answer

a. Proved

b. A solution curve has no relative extrema.

c. Proved

d. The graph is drawn below

Step-by-step solution

Definition of Ordinary Differential Equation

A system of ordinary differential equations is two or more equations involving the derivatives of two or more unknown functions of a single independent variable.

Compute Differentiation

a.

Given Suppose $y\text{'}=y^2+4.$is a constant solution.

Then we have both that $y\text{'}=c\text{'}=0$and $y\text{'}=c^2+4>0$, a contradiction.

Therefore there are no constant solutions.

Describe the graph solution

b.

Note that if $y=\varphi \left(x\right)$is a solution we have that $\varphi \text{'}=\varphi (x{)}^2+4>0$, so the graph of$\varnothing$ is monotonously increasing for all x. In particular a solution has no relative extrema.

Therefore, a solution curve has no relative extrema.

Point of inflection of a solution curve

c.

For a solution $y=\varphi \left(x\right)$we have that $\varphi \text{'}\text{'}\left(x\right)={\left(y\text{'}\right)}^{\text{'}}={\left(y^2+4\right)}^{\text{'}}=2y$which changes sign at $y=0$,

Therefore the y-coordinate of a point of inflection of a solution curve is always 0.

Graph for the equation

d.

From the previous items, the graph of a solution $\varphi \left(x\right)$to this DE should be something like the graph bellow:

Hence, this isthe graph of a solution$\mathit{y}\mathbf{=}\mathit{\varphi }\left(x\right)$of the differential equation whose shape is suggested by parts (a) -(c).