Let $f$ be defined on $I = ]-\pi/2, \pi/2[$ by $f(x) = \frac{\sin x + 1}{\cos x}$. Show $$\forall x \in I, \quad 2f^{\prime}(x) = f(x)^2 + 1.$$

Prove that the family of curves
$$\frac{x^{2}}{a^{2} + \lambda} + \frac{y^{2}}{b^{2} + \lambda} = 1$$
satisfies
$$\frac{dy}{dx}\left(a^{2} - b^{2}\right) = \left(x + y\frac{dy}{dx}\right)\left(x\frac{dy}{dx} - y\right).$$

Let $x ^ { k } + y ^ { k } = a ^ { k }$, $(a, k > 0)$ and $\frac { d y } { d x } + \left( \frac { y } { x } \right) ^ { \frac { 1 } { 3 } } = 0$, then $k$ is
(1) $\frac { 3 } { 2 }$
(2) $\frac { 4 } { 3 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 1 } { 3 }$