F5 - Substitution strategies


Example 1 πŸ”—

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. \( 0 = t y {y'} - t^{2} + y^{2} \)
  2. \( -{y'} - \frac{2}{4 \, t + y - 1} = -4 \, t - y - 3 \)

Answer:

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

Example 2 πŸ”—

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

Answer:

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

Example 3 πŸ”—

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

Answer:

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

Example 4 πŸ”—

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

Answer:

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

Example 5 πŸ”—

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. \( 5 \, t + y + \frac{3}{5 \, t + y + 5} + 10 = {y'} \)
  2. \( -t y {y'} - y^{2} = 3 \, t^{2} \)

Answer:

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

Example 6 πŸ”—

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. \( -3 \, t = y - {y'} + \frac{4}{3 \, t + y + 5} + 8 \)
  2. \( t^{2} = -t y {y'} - y^{2} \)

Answer:

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

Example 7 πŸ”—

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. \( -3 \, t^{2} = -t y {y'} - y^{2} \)
  2. \( y - {y'} - \frac{1}{3 \, t + y + 3} = -3 \, t - 6 \)

Answer:

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

Example 8 πŸ”—

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

Answer:

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

Example 9 πŸ”—

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

Answer:

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

Example 10 πŸ”—

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

Answer:

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

Example 11 πŸ”—

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. \( -5 \, t + y = {y'} - \frac{4}{5 \, t - y - 1} + 4 \)
  2. \( t y {y'} + t^{2} + y^{2} = 0 \)

Answer:

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

Example 12 πŸ”—

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

Answer:

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

Example 13 πŸ”—

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

Answer:

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

Example 14 πŸ”—

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

Answer:

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

Example 15 πŸ”—

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

Answer:

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

Example 16 πŸ”—

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

Answer:

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

Example 17 πŸ”—

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

Answer:

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

Example 18 πŸ”—

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

Answer:

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

Example 19 πŸ”—

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

Answer:

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

Example 20 πŸ”—

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. \( -y + {y'} - \frac{2}{5 \, t + y - 4} = 5 \, t + 1 \)
  2. \( -t y {y'} - t^{2} = y^{2} \)

Answer:

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

Example 21 πŸ”—

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

Answer:

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

Example 22 πŸ”—

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. \( y + \frac{3}{4 \, t + y - 1} = -4 \, t + {y'} - 3 \)
  2. \( t y {y'} - 2 \, t^{2} = -y^{2} \)

Answer:

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

Example 23 πŸ”—

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

Answer:

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

Example 24 πŸ”—

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. \( -t = y - {y'} - \frac{2}{t + y - 1} \)
  2. \( t y {y'} - t^{2} + y^{2} = 0 \)

Answer:

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

Example 25 πŸ”—

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

Answer:

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

Example 26 πŸ”—

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

Answer:

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

Example 27 πŸ”—

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. \( 3 \, t + \frac{4}{3 \, t - y - 1} = y - {y'} - 2 \)
  2. \( -y^{2} = t y {y'} + 2 \, t^{2} \)

Answer:

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

Example 28 πŸ”—

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. \( 0 = -t y {y'} - 5 \, t^{2} - y^{2} \)
  2. \( -{y'} + \frac{5}{3 \, t + y + 3} + 6 = -3 \, t - y \)

Answer:

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

Example 29 πŸ”—

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. \( 5 \, t - y + 1 = -{y'} + \frac{3}{5 \, t - y - 4} \)
  2. \( y^{2} = -t y {y'} - t^{2} \)

Answer:

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

Example 30 πŸ”—

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

Answer:

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

Example 31 πŸ”—

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. \( 3 \, t - y = -{y'} - \frac{4}{3 \, t - y - 4} + 1 \)
  2. \( -t y {y'} - 3 \, t^{2} = y^{2} \)

Answer:

  1. \( \frac{u {u'}}{u^{2} + 4} = 1 \)
  2. \( \frac{u {u'}}{u^{2} + 3} = \frac{1}{t} \)

Example 32 πŸ”—

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. \( -{y'} = -4 \, t - y + \frac{1}{4 \, t + y - 4} \)
  2. \( -y^{2} = t y {y'} + 5 \, t^{2} \)

Answer:

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

Example 33 πŸ”—

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

Answer:

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

Example 34 πŸ”—

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. \( -y + {y'} + \frac{1}{3 \, t - y + 1} = -3 \, t - 4 \)
  2. \( t^{2} = -t y {y'} - y^{2} \)

Answer:

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

Example 35 πŸ”—

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

Answer:

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

Example 36 πŸ”—

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. \( {y'} - \frac{1}{t - y - 2} = -t + y + 1 \)
  2. \( -t y {y'} + 3 \, t^{2} - y^{2} = 0 \)

Answer:

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

Example 37 πŸ”—

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. \( t y {y'} - 3 \, t^{2} = -y^{2} \)
  2. \( \frac{1}{3 \, t + y + 1} - 4 = 3 \, t + y - {y'} \)

Answer:

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

Example 38 πŸ”—

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. \( -y^{2} = t y {y'} - 4 \, t^{2} \)
  2. \( y = -3 \, t + {y'} + \frac{5}{3 \, t + y + 3} - 6 \)

Answer:

  1. \( \frac{u {u'}}{u^{2} - 4} = \frac{1}{t} \)
  2. \( \frac{u {u'}}{u^{2} - 5} = 1 \)

Example 39 πŸ”—

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. \( -5 \, t + {y'} = y - \frac{2}{5 \, t + y - 1} + 4 \)
  2. \( -t y {y'} - y^{2} = -5 \, t^{2} \)

Answer:

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

Example 40 πŸ”—

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. \( -t y {y'} - 2 \, t^{2} - y^{2} = 0 \)
  2. \( y - {y'} + \frac{4}{5 \, t + y + 1} + 6 = -5 \, t \)

Answer:

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

Example 41 πŸ”—

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

Answer:

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

Example 42 πŸ”—

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

Answer:

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

Example 43 πŸ”—

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. \( \frac{4}{2 \, t + y - 5} - 3 = -2 \, t - y + {y'} \)
  2. \( 0 = -t y {y'} - t^{2} - y^{2} \)

Answer:

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

Example 44 πŸ”—

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. \( -t y {y'} + 4 \, t^{2} = y^{2} \)
  2. \( -5 \, t + \frac{5}{5 \, t + y - 4} - 1 = y - {y'} \)

Answer:

  1. \( \frac{u {u'}}{u^{2} - 4} = \frac{1}{t} \)
  2. \( \frac{u {u'}}{u^{2} - 5} = 1 \)

Example 45 πŸ”—

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

Answer:

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

Example 46 πŸ”—

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

Answer:

  1. \( \frac{u {u'}}{u^{2} - 4} = \frac{1}{t} \)
  2. \( \frac{u {u'}}{u^{2} - 5} = 1 \)

Example 47 πŸ”—

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

Answer:

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

Example 48 πŸ”—

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. \( {y'} = 3 \, t + y + \frac{2}{3 \, t + y + 1} + 4 \)
  2. \( -5 \, t^{2} = t y {y'} + y^{2} \)

Answer:

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

Example 49 πŸ”—

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. \( -t y {y'} + 2 \, t^{2} - y^{2} = 0 \)
  2. \( -2 \, t - {y'} = -y - \frac{3}{2 \, t - y + 1} + 3 \)

Answer:

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

Example 50 πŸ”—

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. \( -\frac{2}{5 \, t + y + 5} - 10 = 5 \, t + y - {y'} \)
  2. \( -t y {y'} + t^{2} - y^{2} = 0 \)

Answer:

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