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

Answer:

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