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

Answer:

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