(*): \((a_1x+b_1y+c_1)dx+(a_2x+b_2y+c_2)dy=0\) where \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = k \in \mathbb{R}\. Show that (*) can be transformed into a separable ODE by substituting \(z = a_2x + b_2y\). Then solve (**): \((3x-y+1)dx - (6x-2y-3)dy = 0\)
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Step 1: Given the equation (*): (a₁x + b₁y + c₁)dx + (a₂x + b₂y + c₂)dy = 0, where (a₁)/(a₂) = (b₁)/(b₂) = k in R, we can rewrite it as: (a₁ + b₁k)y dx + a₂x dx + (b₂k + c₁)dy + c₂dy = 0 Show more…
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