Apply each of the substitutions \(u=\frac{y}{t}\) and \(u=at+by+c\) to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.)

  1. \( -3 \, t + {y'} = y - \frac{2}{3 \, t + y - 5} - 2 \)
  2. \( -t y {y'} - 2 \, t^{2} - y^{2} = 0 \)

Answer:

  1. \( \frac{u {u'}}{u^{2} - 2} = 1 \)
  2. \( \frac{u {u'}}{u^{2} + 2} = \frac{1}{t} \)