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. \( -t y {y'} - 5 \, t^{2} = y^{2} \)
  2. \( y = -4 \, t + {y'} + \frac{3}{4 \, t + y + 3} - 7 \)

Answer:

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