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. \( -t y {y'} + 2 \, t^{2} = y^{2} \)
  2. \( y - {y'} - 4 = 2 \, t - \frac{4}{2 \, t - y + 2} \)

Answer:

  1. \( \frac{u {u'}}{u^{2} - 2} = \frac{1}{t} \)
  2. \( \frac{u {u'}}{u^{2} - 4} = 1 \)