2. (x csc y/x - y) dx + xdy=0 3. (x^2 + 2xy - 4y^2) dx - ( x^2 - 8xy - 4 y^2)=0 4. x^y ' = 4x^2 + 7xy + 2 y^2
Solution (2) $\left[ x \csc \left( \dfrac{y}{x} \right) - y \right] \, dx + x \, dy = 0$
$\left[ x \csc \left( \dfrac{vx}{x} \right) - vx \right] \, dx + x(v \, dx + x \, dv) = 0$
$(x \csc v - vx) \, dx + vx \, dx + x^2 \, dv = 0$
$x \csc v \, dx + x^2 \, dv = 0$
$\dfrac{dx}{x} + \dfrac{dv}{\csc v} = 0$
$\dfrac{dx}{x} + \sin v \, dv = 0$
$\ln x - \cos v = c$
$\ln x - \cos \left(\dfrac{y}{x} \right) = c$
thanks po sa solution.. yung prob. 3 and4. po...solution.
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Solution (2)
$\left[ x \csc \left( \dfrac{y}{x} \right) - y \right] \, dx + x \, dy = 0$
dy = v dx + x dv
$\left[ x \csc \left( \dfrac{vx}{x} \right) - vx \right] \, dx + x(v \, dx + x \, dv) = 0$
$(x \csc v - vx) \, dx + vx \, dx + x^2 \, dv = 0$
$x \csc v \, dx + x^2 \, dv = 0$
$\dfrac{dx}{x} + \dfrac{dv}{\csc v} = 0$
$\dfrac{dx}{x} + \sin v \, dv = 0$
$\ln x - \cos v = c$
$\ln x - \cos \left(\dfrac{y}{x} \right) = c$
thanks po sa solution.. yung prob. 3 and4. po...solution.
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