## F2 - Separation of variables

#### Example 1 π

Find the solution to the given IVP.

$-\frac{6}{{y}^{2}} = -2 \, {y} {y'} + \frac{4 \, t}{{y}^{2}} \hspace{2em} y( 3 )= 2$

${y} = {\left(t^{2} + 3 \, t - 10\right)}^{\frac{1}{3}}$

#### Example 2 π

Find the solution to the given IVP.

$-3 \, {y} \cos\left(t\right) + 3 \, {y'} = 0 \hspace{2em} y( 0 )= -4$

${y} = -4 \, e^{\sin\left(t\right)}$

#### Example 3 π

Find the solution to the given IVP.

$-\frac{12 \, t}{{y}} - \frac{6}{{y}} = -3 \, {y} {y'} \hspace{2em} y( 2 )= 3$

${y} = \sqrt{2 \, t^{2} + 2 \, t - 3}$

#### Example 4 π

Find the solution to the given IVP.

$6 \, {y} = 2 \, {y'} t \hspace{2em} y( -1 )= -1$

${y} = t^{3}$

#### Example 5 π

Find the solution to the given IVP.

$0 = -2 \, {y'} t - 6 \, {y} \hspace{2em} y( -2 )= -\frac{1}{4}$

${y} = \frac{2}{t^{3}}$

#### Example 6 π

Find the solution to the given IVP.

$0 = -2 \, {y'} t + 4 \, {y} \hspace{2em} y( 3 )= 27$

${y} = 3 \, t^{2}$

#### Example 7 π

Find the solution to the given IVP.

$2 \, {y} {y'} - \frac{12 \, t}{{y}} - \frac{2}{{y}} = 0 \hspace{2em} y( 2 )= 4$

${y} = \sqrt{3 \, t^{2} + t + 2}$

#### Example 8 π

Find the solution to the given IVP.

$-4 \, {y} = 2 \, {y'} t \hspace{2em} y( 2 )= \frac{1}{4}$

${y} = \frac{1}{t^{2}}$

#### Example 9 π

Find the solution to the given IVP.

$-2 \, {y'} = -2 \, {y} \cos\left(t\right) \hspace{2em} y( 0 )= -1$

${y} = -e^{\sin\left(t\right)}$

#### Example 10 π

Find the solution to the given IVP.

$-3 \, {y} \sin\left(t\right) = 3 \, {y'} \hspace{2em} y( 0 )= 4 \, e$

${y} = 4 \, e^{\cos\left(t\right)}$

#### Example 11 π

Find the solution to the given IVP.

$6 \, {y} {\left(t - 1\right)} = 3 \, {y'} \hspace{2em} y( 0 )= 2 \, e^{\left(-5\right)}$

${y} = 2 \, e^{\left(t^{2} - 2 \, t - 5\right)}$

#### Example 12 π

Find the solution to the given IVP.

$9 \, {y} = -3 \, {y'} t \hspace{2em} y( -2 )= -\frac{1}{8}$

${y} = \frac{1}{t^{3}}$

#### Example 13 π

Find the solution to the given IVP.

$3 \, {y'} t + 6 \, {y} = 0 \hspace{2em} y( 2 )= -\frac{1}{2}$

${y} = -\frac{2}{t^{2}}$

#### Example 14 π

Find the solution to the given IVP.

$-4 \, {y} \sin\left(t\right) - 2 \, {y'} = 0 \hspace{2em} y( 0 )= 3 \, e^{2}$

${y} = 3 \, e^{\left(2 \, \cos\left(t\right)\right)}$

#### Example 15 π

Find the solution to the given IVP.

$-2 \, {y'} {y} = -\frac{12 \, t}{{y}^{2}} - \frac{2}{{y}^{2}} \hspace{2em} y( -2 )= 3$

${y} = {\left(3 \, t^{2} + t + 17\right)}^{\frac{1}{3}}$

#### Example 16 π

Find the solution to the given IVP.

$3 \, {y'} t = 6 \, {y} \hspace{2em} y( 1 )= -2$

${y} = -2 \, t^{2}$

#### Example 17 π

Find the solution to the given IVP.

$-2 \, {y'} = 4 \, {y} {\left(t + 1\right)} \hspace{2em} y( 0 )= -e^{2}$

${y} = -e^{\left(-t^{2} - 2 \, t + 2\right)}$

#### Example 18 π

Find the solution to the given IVP.

$0 = -2 \, {y} {y'} + \frac{12 \, t}{{y}} + \frac{2}{{y}} \hspace{2em} y( -2 )= 4$

${y} = \sqrt{3 \, t^{2} + t + 6}$

#### Example 19 π

Find the solution to the given IVP.

$-2 \, {y'} t - 4 \, {y} = 0 \hspace{2em} y( -2 )= -\frac{3}{4}$

${y} = -\frac{3}{t^{2}}$

#### Example 20 π

Find the solution to the given IVP.

$0 = -3 \, {y'} t - 9 \, {y} \hspace{2em} y( 3 )= \frac{1}{27}$

${y} = \frac{1}{t^{3}}$

#### Example 21 π

Find the solution to the given IVP.

$-2 \, {y'} t - 6 \, {y} = 0 \hspace{2em} y( 4 )= -\frac{3}{64}$

${y} = -\frac{3}{t^{3}}$

#### Example 22 π

Find the solution to the given IVP.

$-3 \, {y} \cos\left(t\right) = -3 \, {y'} \hspace{2em} y( 0 )= -3$

${y} = -3 \, e^{\sin\left(t\right)}$

#### Example 23 π

Find the solution to the given IVP.

$-12 \, {y} t = -3 \, {y'} \hspace{2em} y( 0 )= -4$

${y} = -4 \, e^{\left(2 \, t^{2}\right)}$

#### Example 24 π

Find the solution to the given IVP.

$0 = -2 \, {y'} t + 6 \, {y} \hspace{2em} y( 2 )= 32$

${y} = 4 \, t^{3}$

#### Example 25 π

Find the solution to the given IVP.

$0 = 3 \, {y'} t + 9 \, {y} \hspace{2em} y( 4 )= \frac{1}{32}$

${y} = \frac{2}{t^{3}}$

#### Example 26 π

Find the solution to the given IVP.

$-6 \, {y} \cos\left(t\right) - 3 \, {y'} = 0 \hspace{2em} y( 0 )= 3$

${y} = 3 \, e^{\left(-2 \, \sin\left(t\right)\right)}$

#### Example 27 π

Find the solution to the given IVP.

$\frac{12 \, t}{{y}} + \frac{4}{{y}} = 2 \, {y} {y'} \hspace{2em} y( -3 )= 2$

${y} = \sqrt{3 \, t^{2} + 2 \, t - 17}$

#### Example 28 π

Find the solution to the given IVP.

$0 = -6 \, {y} \cos\left(t\right) + 2 \, {y'} \hspace{2em} y( 0 )= -4$

${y} = -4 \, e^{\left(3 \, \sin\left(t\right)\right)}$

#### Example 29 π

Find the solution to the given IVP.

$0 = 2 \, {y'} t + 6 \, {y} \hspace{2em} y( -3 )= \frac{4}{27}$

${y} = -\frac{4}{t^{3}}$

#### Example 30 π

Find the solution to the given IVP.

$-3 \, {\left(2 \, t - 1\right)} {y} = -3 \, {y'} \hspace{2em} y( 0 )= 4 \, e^{3}$

${y} = 4 \, e^{\left(t^{2} - t + 3\right)}$

#### Example 31 π

Find the solution to the given IVP.

$-\frac{12 \, t}{{y}} - \frac{6}{{y}} = -2 \, {y}^{2} {y'} \hspace{2em} y( -2 )= 2$

${y} = {\left(3 \, t^{2} + 3 \, t + 2\right)}^{\frac{1}{3}}$

#### Example 32 π

Find the solution to the given IVP.

$0 = 6 \, {y} \cos\left(t\right) - 2 \, {y'} \hspace{2em} y( 0 )= -4$

${y} = -4 \, e^{\left(3 \, \sin\left(t\right)\right)}$

#### Example 33 π

Find the solution to the given IVP.

$3 \, {y'} = 3 \, {y} {\left(6 \, t + 1\right)} \hspace{2em} y( 0 )= 3 \, e^{\left(-3\right)}$

${y} = 3 \, e^{\left(3 \, t^{2} + t - 3\right)}$

#### Example 34 π

Find the solution to the given IVP.

$-2 \, {y} {\left(6 \, t - 1\right)} = -2 \, {y'} \hspace{2em} y( 0 )= -e^{4}$

${y} = -e^{\left(3 \, t^{2} - t + 4\right)}$

#### Example 35 π

Find the solution to the given IVP.

$0 = 2 \, {y} \cos\left(t\right) + 2 \, {y'} \hspace{2em} y( 0 )= 2$

${y} = 2 \, e^{\left(-\sin\left(t\right)\right)}$

#### Example 36 π

Find the solution to the given IVP.

$0 = 9 \, {y} \sin\left(t\right) + 3 \, {y'} \hspace{2em} y( 0 )= e^{3}$

${y} = e^{\left(3 \, \cos\left(t\right)\right)}$

#### Example 37 π

Find the solution to the given IVP.

$0 = -6 \, {y} \cos\left(t\right) - 2 \, {y'} \hspace{2em} y( 0 )= 2$

${y} = 2 \, e^{\left(-3 \, \sin\left(t\right)\right)}$

#### Example 38 π

Find the solution to the given IVP.

$0 = 3 \, {y'} {y} - \frac{12 \, t}{{y}} - \frac{6}{{y}} \hspace{2em} y( 2 )= 2$

${y} = \sqrt{2 \, t^{2} + 2 \, t - 8}$

#### Example 39 π

Find the solution to the given IVP.

$-3 \, {y'} t + 9 \, {y} = 0 \hspace{2em} y( -2 )= 32$

${y} = -4 \, t^{3}$

#### Example 40 π

Find the solution to the given IVP.

$3 \, {y}^{2} {y'} - \frac{9}{{y}} = \frac{12 \, t}{{y}} \hspace{2em} y( -2 )= 3$

${y} = {\left(2 \, t^{2} + 3 \, t + 25\right)}^{\frac{1}{3}}$

#### Example 41 π

Find the solution to the given IVP.

$-2 \, {\left(2 \, t + 3\right)} {y} + 2 \, {y'} = 0 \hspace{2em} y( 0 )= -4 \, e^{\left(-1\right)}$

${y} = -4 \, e^{\left(t^{2} + 3 \, t - 1\right)}$

#### Example 42 π

Find the solution to the given IVP.

$-2 \, {y'} = -2 \, {y} \sin\left(t\right) \hspace{2em} y( 0 )= -4 \, e^{\left(-1\right)}$

${y} = -4 \, e^{\left(-\cos\left(t\right)\right)}$

#### Example 43 π

Find the solution to the given IVP.

$-9 \, {y} \sin\left(t\right) = -3 \, {y'} \hspace{2em} y( 0 )= -e^{\left(-3\right)}$

${y} = -e^{\left(-3 \, \cos\left(t\right)\right)}$

#### Example 44 π

Find the solution to the given IVP.

$-\frac{4 \, t}{{y}} - \frac{2}{{y}} = -2 \, {y} {y'} \hspace{2em} y( 2 )= 3$

${y} = \sqrt{t^{2} + t + 3}$

#### Example 45 π

Find the solution to the given IVP.

$-3 \, {y'} t + 9 \, {y} = 0 \hspace{2em} y( 4 )= -64$

${y} = -t^{3}$

#### Example 46 π

Find the solution to the given IVP.

$0 = 3 \, {y'} t + 6 \, {y} \hspace{2em} y( 4 )= \frac{1}{8}$

${y} = \frac{2}{t^{2}}$

#### Example 47 π

Find the solution to the given IVP.

$-6 \, {y} = 3 \, {y'} t \hspace{2em} y( 1 )= 4$

${y} = \frac{4}{t^{2}}$

#### Example 48 π

Find the solution to the given IVP.

$6 \, {y} \cos\left(t\right) + 2 \, {y'} = 0 \hspace{2em} y( 0 )= 1$

${y} = e^{\left(-3 \, \sin\left(t\right)\right)}$

#### Example 49 π

Find the solution to the given IVP.

$0 = 2 \, t {y'} + 4 \, {y} \hspace{2em} y( 1 )= 2$

${y} = \frac{2}{t^{2}}$
$0 = 3 \, {y'} t - 9 \, {y} \hspace{2em} y( 2 )= 32$
${y} = 4 \, t^{3}$