## F5 - Substitution strategies

#### Example 1 🔗

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

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

#### Example 2 🔗

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. $$t y {y'} = -5 \, t^{2} - y^{2}$$

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

#### Example 3 🔗

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. $$t + {y'} - 1 = y - \frac{2}{t - y - 2}$$

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

#### Example 4 🔗

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. $$y = -t + {y'} - \frac{4}{t + y - 4} + 3$$

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

#### Example 5 🔗

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. $$-t y {y'} - y^{2} = 3 \, t^{2}$$

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

#### Example 6 🔗

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. $$t^{2} = -t y {y'} - y^{2}$$

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

#### Example 7 🔗

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. $$y - {y'} - \frac{1}{3 \, t + y + 3} = -3 \, t - 6$$

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

#### Example 8 🔗

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. $$0 = -t y {y'} - 4 \, t^{2} - y^{2}$$

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

#### Example 9 🔗

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. $$4 \, t^{2} = -t y {y'} - y^{2}$$

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

#### Example 10 🔗

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. $$-3 \, t^{2} - y^{2} = t y {y'}$$

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

#### Example 11 🔗

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. $$t y {y'} + t^{2} + y^{2} = 0$$

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

#### Example 12 🔗

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. $$0 = 5 \, t + y - {y'} - \frac{2}{5 \, t + y - 5}$$

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

#### Example 13 🔗

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. $$3 \, t + {y'} - \frac{2}{3 \, t - y - 3} = y$$

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

#### Example 14 🔗

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. $$5 \, t + y - {y'} = \frac{4}{5 \, t + y - 5}$$

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

#### Example 15 🔗

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

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

#### Example 16 🔗

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

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

#### Example 17 🔗

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. $$-{y'} = 2 \, t - y + \frac{2}{2 \, t - y - 5} - 3$$

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

#### Example 18 🔗

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. $$0 = t y {y'} + t^{2} + y^{2}$$

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

#### Example 19 🔗

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. $$-t + y - {y'} + 2 = \frac{3}{t - y - 3}$$

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

#### Example 20 🔗

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. $$-t y {y'} - t^{2} = y^{2}$$

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

#### Example 21 🔗

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. $$-5 \, t - y + {y'} - 9 = \frac{2}{5 \, t + y + 4}$$

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

#### Example 22 🔗

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. $$t y {y'} - 2 \, t^{2} = -y^{2}$$

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

#### Example 23 🔗

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. $$2 \, t + {y'} = y + \frac{5}{2 \, t - y - 2}$$

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

#### Example 24 🔗

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. $$t y {y'} - t^{2} + y^{2} = 0$$

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

#### Example 25 🔗

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. $$-t y {y'} = -5 \, t^{2} + y^{2}$$

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

#### Example 26 🔗

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. $$3 \, t + y - {y'} = -\frac{1}{3 \, t + y - 1} - 2$$

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

#### Example 27 🔗

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. $$-y^{2} = t y {y'} + 2 \, t^{2}$$

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

#### Example 28 🔗

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. $$-{y'} + \frac{5}{3 \, t + y + 3} + 6 = -3 \, t - y$$

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

#### Example 29 🔗

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. $$y^{2} = -t y {y'} - t^{2}$$

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

#### Example 30 🔗

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. $$y = -t + {y'} - \frac{4}{t + y - 3} + 2$$

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

#### Example 31 🔗

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. $$-t y {y'} - 3 \, t^{2} = y^{2}$$

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

#### Example 32 🔗

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. $$-y^{2} = t y {y'} + 5 \, t^{2}$$

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

#### Example 33 🔗

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. $$-t y {y'} - t^{2} = y^{2}$$

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

#### Example 34 🔗

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. $$t^{2} = -t y {y'} - y^{2}$$

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

#### Example 35 🔗

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. $$0 = -4 \, t + y - {y'} - \frac{2}{4 \, t - y - 4}$$

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

#### Example 36 🔗

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. $$-t y {y'} + 3 \, t^{2} - y^{2} = 0$$

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

#### Example 37 🔗

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. $$\frac{1}{3 \, t + y + 1} - 4 = 3 \, t + y - {y'}$$

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

#### Example 38 🔗

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. $$y = -3 \, t + {y'} + \frac{5}{3 \, t + y + 3} - 6$$

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

#### Example 39 🔗

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. $$-t y {y'} - y^{2} = -5 \, t^{2}$$

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

#### Example 40 🔗

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. $$y - {y'} + \frac{4}{5 \, t + y + 1} + 6 = -5 \, t$$

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

#### Example 41 🔗

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. $$5 \, t^{2} - y^{2} = t y {y'}$$

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

#### Example 42 🔗

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. $$t + y - {y'} + 4 = -\frac{5}{t + y + 3}$$

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

#### Example 43 🔗

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. $$0 = -t y {y'} - t^{2} - y^{2}$$

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

#### Example 44 🔗

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. $$-5 \, t + \frac{5}{5 \, t + y - 4} - 1 = y - {y'}$$

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

#### Example 45 🔗

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. $$-t - y = -{y'} - \frac{3}{t + y - 1}$$

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

#### Example 46 🔗

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. $$2 \, t + y - \frac{5}{2 \, t + y + 5} + 7 = {y'}$$

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

#### Example 47 🔗

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. $$-{y'} - \frac{1}{5 \, t - y - 3} - 2 = 5 \, t - y$$

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

#### Example 48 🔗

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. $$-5 \, t^{2} = t y {y'} + y^{2}$$

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

#### Example 49 🔗

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. $$-2 \, t - {y'} = -y - \frac{3}{2 \, t - y + 1} + 3$$

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

#### Example 50 🔗

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. $$-t y {y'} + t^{2} - y^{2} = 0$$

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