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 four parts such that each part passes the vertical test.

Step-by-step solution

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 given ellipse is shown blue.

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 left grey dashed vertical line intersects the curve once and the right dashed vertical line intersects the curve twice, this will be out boundary lines.

Therefore,

We can divide the curve into four parts that pass the vertical test:

${\varphi }_1$: The part when$-\infty <x<0$(outside the loop)

${\varphi }_2$: The part of the loop above the blue dashed line.

${\varphi }_3$: The part of the loop below the blue dashed line.

${\varphi }_4$: The part when $o⩽x<\infty$(outside the loop).