Questions: Assinale, qual das EDO abaixo é exata (A) (1+x^2 y^2) dx+(x^2 y^2) dy=0 (B) (ln y+x^2) dx+(y e^x) dy=0 (C) (xy2) dx+(tg(x)-y) dy=0 (D) (ln y+sec (x)) dx+(y^3-e^x) dy=0 (E) (y cos (x)) dx+(sin (x)+y^2) dy=0

Transcript text: Assinale, qual das EDO abaixo é exata (A) $\left(1+x^{2} y^{2}\right) d x+\left(x^{2} y^{2}\right) d y=0$ (B) $\left(\ln |y|+x^{2}\right) d x+\left(y e^{x}\right) d y=0$ (C) $(x y 2) d x+(\operatorname{tg}(x)-y) d y=0$ (D) $(\ln |y|+\sec (x)) d x+\left(y^{3}-e^{x}\right) d y=0$ (E) $(y \cos (x)) d x+\left(\sin (x)+y^{2}\right) d y=0$

Solution

Solution Steps

To determine which of the given differential equations (EDOs) is exact, we need to check if the mixed partial derivatives of the functions multiplying $dx$ and $dy$ are equal. Specifically, for an equation of the form $M(x, y) dx + N(x, y) dy = 0$, the equation is exact if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.

Solution Approach

Extract the functions $M(x, y)$ and $N(x, y)$ from each option.
Compute the partial derivative of $M$ with respect to $y$ and the partial derivative of $N$ with respect to $x$.
Compare the partial derivatives for each option to determine if they are equal.

Step 1: Identificação das Funções

Para cada uma das equações diferenciais apresentadas, identificamos as funções $M(x, y)$ e $N(x, y)$:

(A) $M = 1 + x^2 y^2$, $N = x^2 y^2$
(B) $M = \ln |y| + x^2$, $N = y e^x$
(C) $M = x y^2$, $N = \tan(x) - y$
(D) $M = \ln |y| + \sec(x)$, $N = y^3 - e^x$
(E) $M = y \cos(x)$, $N = \sin(x) + y^2$

Step 2: Cálculo das Derivadas Parciais

Calculamos as derivadas parciais de $M$ em relação a $y$ e de $N$ em relação a $x$ para cada opção:

(A) $\frac{\partial M}{\partial y} = 2x^2 y$ e $\frac{\partial N}{\partial x} = 0$
(B) $\frac{\partial M}{\partial y} = \frac{1}{y}$ e $\frac{\partial N}{\partial x} = y e^x$
(C) $\frac{\partial M}{\partial y} = 2xy$ e $\frac{\partial N}{\partial x} = \sec^2(x)$
(D) $\frac{\partial M}{\partial y} = \frac{1}{y}$ e $\frac{\partial N}{\partial x} = -e^x$
(E) $\frac{\partial M}{\partial y} = \cos(x)$ e $\frac{\partial N}{\partial x} = \cos(x)$

Step 3: Verificação da Exatidão

Verificamos se as derivadas parciais são iguais: