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. \( -t - y + {y'} = -\frac{2}{t + y + 2} + 3 \)
  2. \( t y {y'} - 4 \, t^{2} + y^{2} = 0 \)

Answer:

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