12- Resolva o problema de valor inicial \( y^{\prime \prime}+ \) \( 8 y^{\prime}-9 y=0 ; y(1)=1 ; y^{\prime}(1)=0 \)

Find \( y^{\prime} \) given \( y=x^{2} \cos x \)

Find the Laplace transform \( F(s)=\mathcal{L}\{f(t)\} \) of the function \( f(t)=(7-t)(h(t-2)-h(t-6) \), for \( s \neq 0 \). \( F(s)=\mathcal{L}\{f(t)\}=1 \) help (formulas)

2. (4 pts) What is the derivative of \( y=x^{\pi}+e^{3}+\ln \left(3^{x}\right)+\pi^{x} \). Be sure to answer with a derivative notation. \[ \text { ANSWER: } \quad y^{\prime}=\pi x^{\pi-1}+\ln (3)+\pi^{x} \ln (\pi) \]

\( \left. \begin{array} { l } { \int 5 d x = } \\ { \int \frac { d x } { 3 } = } \\ { \int + d x = } \\ { \int 3 d u = } \end{array} \right. \)

25. \( N=\frac{p+q}{p+r}, \quad p=u+v w, \quad q=v+u w, \quad r=w+u v \) \( \frac{\partial N}{\partial u}, \frac{\partial N}{\partial v}, \frac{\partial N}{\partial w} \quad \) donde \( u=2, v=3, w=4 \)