Warning: this is an htmlized version!
The original is across this link,
and the conversion rules are here.
% (find-angg "LATEX/2017-1-C2-P2.tex")
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%               file:///tmp/2017-1-C2-P2.pdf
%           file:///tmp/pen/2017-1-C2-P2.pdf
% http://angg.twu.net/LATEX/2017-1-C2-P2.pdf
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%
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%
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%
\begin{document}

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{\setlength{\parindent}{0em}
\footnotesize
\par CÃˇlculo 2
\par PURO-UFF - 2017.1
\par P2 - 18/jul/2017 - Eduardo Ochs
\par Respostas sem justificativas nÃŁo serÃŁo aceitas.
\par Proibido usar quaisquer aparelhos eletrÃ∧nicos.

}

\bsk
\bsk

\setlength{\parindent}{0em}
\def\T(Total: #1 pts){{\bf(Total: #1 pts)}}
\def\T(Total: #1 pts){{\bf(Total: #1)}}
\def\B       (#1 pts){{\bf(#1 pts)}}
% Usage:
% 1) \T(Total: 2.34 pts) Foo
% a) \B(0.45 pts) Bar








% (find-angg "LATEX/2015-2-GA-P2.tex")

1) \T(Total: 2.0 pts) Seja $(*)$ a seguinte EDO: $f''-5f+6f=0$.

a) \B(0.5 pts) Expresse $(*)$ na forma $(D-a)(D-b)f=0$.

b) \B(0.5 pts) Encontre as soluções bÃˇsicas de $(*)$.

c) \B(0.2 pts) Encontre uma soluÃ§ÃŁo de $(*)$ que obedeça $f(0)=1$, $f'(0)=0$.

d) \B(0.3 pts) Encontre uma soluÃ§ÃŁo de $(*)$ que obedeça $f(0)=0$, $f'(0)=1$.

e) \B(0.5 pts) Encontre uma soluÃ§ÃŁo de $(*)$ que obedeça $f(0)=2$, $f'(0)=3$.

\bsk
\bsk

2) \T(Total: 3.5 pts) Seja $(**)$ a seguinte EDO: $f''+4f'+13f=0$.

a) \B(1.0 pts) Expresse $(**)$ na forma $(D-a)(D-b)f=0$.

b) \B(1.0 pts) Encontre as soluções bÃˇsicas de $(**)$.

c) \B(1.0 pts) Encontre as soluções {\sl reais} de $(**)$.

d) \B(0.5 pts) Teste as soluções que você encontrou no item anterior.

\bsk
\bsk

3) \T(Total: 2.5 pts) Seja $(***)$ a seguinte EDO: $\ddx y = x e^{-y}$.

a) \B(1.5 pts) Encontre a soluÃ§ÃŁo geral de $(***)$.

b) \B(1.0 pts) Encontre uma soluÃ§ÃŁo de $(***)$ que passa pelo ponto $(3,4)$.

\bsk
\bsk

4) \T(Total: 2.0 pts) Seja $(****)$ a seguinte EDO: $-3x^2 dx + (2y+2) dy = 0$.

a) \B(0.5 pts) Verifique que $(****)$ é exata.

b) \B(1.0 pts) Encontre a soluÃ§ÃŁo geral de $(****)$.

c) \B(0.5 pts) Encontre uma soluÃ§ÃŁo de $(****)$ que passa pelo ponto $(3,4)$.



\newpage

{\bf Gabarito:} (nÃŁo revisado)

\bsk

% (find-es "ipython" "2017.1-C2-P2" "Questao 1")

1a) $(D-2)(D-3)f = 0$

1b) $f_1 = e^{2x}$, $f_2 = e^{3x}$

1c) $3f_1 - 2f_2$

1d) $-f_1 + f_2$

1e) $3f_1 - f_2$

\bsk

% (find-es "ipython" "2017.1-C2-P2" "Questao 2")

2a) $(D-(-2+3i))(D-(-2-3i))f=0$

2b) $f_1 = e^{(-2+3i)x}$,
    $f_2 = e^{(-2-3i)x}$.

2c) $f_3 = \cos(3x)·e^{-2x}$,
    $f_4 = \sen(3x)·e^{-2x}$.

2d)

\bsk

% (find-es "ipython" "2017.1-C2-P2" "Questao 3")

3a) $f = \ln(\frac{x^2}{2} + C)$

3b) $f = \ln(\frac{x^2}{2} - \frac92 + e^4)$

\bsk

4a) $G = -3x^2$, $H=2y+2$, $G_y=0=H_x$; $Gdx + Hdy=0$ é exata, e
existe $F$ tal que $F_x = -3x^2$ e $F_y = 2y+2$.

4b) $F(x,y) = -x^3 +y^2 + 2y$ ou
    $F(x,y) = -x^3 + y^2 + 2y + 1 = -x^3 + (y+1)^2$;

 $F(x,y) = C \quad⇒\quad (y+1)^2 = C+x^3 \quad⇒\quad y = \sqrt{C+x^3}-1$

4c) $4 = \sqrt{C+3^3}-1 \quad⇒\quad \sqrt{C+3^3}=5 \quad⇒\quad
C+3^3=25$

$⇒\quad C=-2 \quad⇒\quad y = \sqrt{-2+x^3}-1$


 
% f2 = sin(3*x) * exp(-2*x)








\end{document}

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