Question

In Problems 49 and 50 the given figure represents the graph of an implicit solution$\mathit{G}\mathbf{(}\mathit{x}\mathbf{,}\mathit{y}\mathbf{)}\mathbf{=}\mathbf{0}$of a differential equation$\mathit{d}\mathit{y}\mathbf{/}\mathit{d}\mathit{x}\mathbf{=}\mathit{f}\mathbf{(}\mathit{x}\mathbf{,}\mathit{y}\mathbf{)}\mathbf{.}$ In each case the relation$\mathit{G}\mathbf{(}\mathit{x}\mathbf{,}\mathit{y}\mathbf{)}\mathbf{=}\mathbf{0}$ implicitly defines several solutions of the DE. Carefully reproduce each figure on a piece of paper. Use different colored pencils to mark off segments, or pieces, on each graph that correspond to graphs of solutions. Keep in mind that a solution$\mathit{\varphi }$ must be a function and differentiable. Use the solution curve to estimate an intervalI of definition of each solution$\mathit{\varphi }$.

Short answer

Divide the curve into two parts such that both parts pass the vertical test.

Step-by-step solution

Step 1:Definition of differential equation.

An equation containing the derivatives of one or more unknown functions (or dependent variables), with respect to one or more independent variables, is said to be a differential equation (DE).

Vertical line.

The sketch of the ellipse is shown blue in the given graph.

Since the solution must be a function, it must satisfy the vertical test, meaning that a vertical line must not intersect the portion of a curve more than once.

The leftmost and the rightmost vertical line intersect the curve only once, all other in between intersect the curve twice.

This means that we can divide the curve into two parts.

${\varphi }_1$: the part above the line $a.$

${\varphi }_2$: The part below the line $a.$

Divide the curve into two parts such that both parts pass the vertical test.