La ecuación de la recta tangente a la curva \( y=\sqrt{x^{2}+5} \) en el punto \( x=-2 \)

\( f(x)=\frac{x^{2}-4 x+7}{1-x} \) demantre que \( f^{\prime}(x)=\frac{4-(x-1)^{2}}{(x-1)^{2}} \) pan \( x \in \mathbb{R}-\{1\} \).

\( y ^ { \prime \prime } - 2 y ^ { \prime } + y = 0 , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = \)

We define the function \( f(x)=\frac{\sin (3 x)}{x} \), where \( \sin (\theta) \) applies to \( \theta \) given in radians. (a) Which of the following best describes the value of \( f(0) \) ? (No answer given)

a) Consider the equation \[ \frac{2 t y}{t^{2}+1}-2 t-\left(2-\ln \left(t^{2}+1\right)\right) y^{\prime}=0 \] i. Determine whether the differential equation is exact. ii. Find every solution of the equation above. ( 3 Marks) Find the general solution to the following Bernoulli Equation \( y^{\prime}+x y=x y^{3} \). ( 6 Marks) c) Find the solution to the following homogeneous differential equation. (5 Marks) \[ \left(4 x^{4}-x^{3} y+y^{4}\right) \mathrm{d} x+x^{4} \mathrm{~d} y=0 . \]

a) Consider the equation \[ \frac{2 t y}{t^{2}+1}-2 t-\left(2-\ln \left(t^{2}+1\right)\right) y^{\prime}=0 \] i. Determine whether the differential equation is exact. ii. Find every solution of the equation above. ( 6 Marks) b) Find the general solution to the following Bernoulli Equation \( y^{\prime}+x y=x y^{3} \). ( 6 Marks) Find the solution to the following homogeneous differential equation. ( 5 Marks)