\( e = \frac { 2 } { \theta + 1 } . \quad \frac { d e } { d \theta } = - \frac { 2 } { ( \theta + 1 ) ^ { 2 } } \)

(2) Nontrer que la forme implicite de la solution de l'ED \( 9 y y^{\prime}+4 x=0 \) est \( \frac{x^{2}}{9}+\frac{y^{2}}{4}=k \) (cste)

Consider the differential equation \( \frac{dy}{dx} = 4x^2 \) with the initial condition \( y(1) = 5 \). Determine the specific solution.

Evaluate the derivative \( y=\sinh (x) \)

1. State whether the following statements are True or False. Justify your answer with the help of a short proof or a counter-example. No marks are awarded for a question without justification : \[ 2 \times 5=10 \] (i) The function \( f(x, y)=x y^{2} \quad \) satisfies Lipschitz condition on : \[ a \leq x \leq b,-\infty<y<\infty \]

\( \lim _ { x \rightarrow 0 } \frac { f ( x ) + \sin x - \sqrt { x + 1 } } { x } = \frac { 3 } { 2 } \)