We study the differential equation $$y(x) y'(x) = -4x \tag{E}$$ We denote $\mathscr{C}$ the image in $\mathbb{R}^2$ of the application $$\gamma : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}^2, t \mapsto (\cos t, 2\sin t)$$ We denote $g$ the function of a real variable with real values whose graph is $\gamma\left(\left[\frac{\pi}{4}, \pi\right]\right)$. II.B.1) Determine the domain of definition $\Delta$ of $g$, as well as an expression for $g$. II.B.2) Verify that the restriction of $g$ to the largest open interval contained in $\Delta$ is a solution of $(E)$. II.B.3) Is this a maximal solution? If not, determine a maximal solution $m$ whose graph includes that of $g$.
We study the differential equation
$$y(x) y'(x) = -4x \tag{E}$$
We denote $\mathscr{C}$ the image in $\mathbb{R}^2$ of the application
$$\gamma : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}^2, t \mapsto (\cos t, 2\sin t)$$
We denote $g$ the function of a real variable with real values whose graph is $\gamma\left(\left[\frac{\pi}{4}, \pi\right]\right)$.
II.B.1) Determine the domain of definition $\Delta$ of $g$, as well as an expression for $g$.\\
II.B.2) Verify that the restriction of $g$ to the largest open interval contained in $\Delta$ is a solution of $(E)$.\\
II.B.3) Is this a maximal solution? If not, determine a maximal solution $m$ whose graph includes that of $g$.