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. \( t y {y'} - 4 \, t^{2} + y^{2} = 0 \)
  2. \( y - \frac{4}{4 \, t - y + 5} - 9 = 4 \, t + {y'} \)

Answer:

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