F2 - Separation of variables


Example 1 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 2 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 3 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 4 πŸ”—

Find the solution to the given IVP.

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

Answer:

\[ {y} = t^{3} \]


Example 5 πŸ”—

Find the solution to the given IVP.

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

Answer:

\[ {y} = \frac{2}{t^{3}} \]


Example 6 πŸ”—

Find the solution to the given IVP.

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

Answer:

\[ {y} = 3 \, t^{2} \]


Example 7 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 8 πŸ”—

Find the solution to the given IVP.

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

Answer:

\[ {y} = \frac{1}{t^{2}} \]


Example 9 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 10 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 11 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 12 πŸ”—

Find the solution to the given IVP.

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

Answer:

\[ {y} = \frac{1}{t^{3}} \]


Example 13 πŸ”—

Find the solution to the given IVP.

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

Answer:

\[ {y} = -\frac{2}{t^{2}} \]


Example 14 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 15 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 16 πŸ”—

Find the solution to the given IVP.

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

Answer:

\[ {y} = -2 \, t^{2} \]


Example 17 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 18 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 19 πŸ”—

Find the solution to the given IVP.

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

Answer:

\[ {y} = -\frac{3}{t^{2}} \]


Example 20 πŸ”—

Find the solution to the given IVP.

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

Answer:

\[ {y} = \frac{1}{t^{3}} \]


Example 21 πŸ”—

Find the solution to the given IVP.

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

Answer:

\[ {y} = -\frac{3}{t^{3}} \]


Example 22 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 23 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 24 πŸ”—

Find the solution to the given IVP.

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

Answer:

\[ {y} = 4 \, t^{3} \]


Example 25 πŸ”—

Find the solution to the given IVP.

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

Answer:

\[ {y} = \frac{2}{t^{3}} \]


Example 26 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 27 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 28 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 29 πŸ”—

Find the solution to the given IVP.

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

Answer:

\[ {y} = -\frac{4}{t^{3}} \]


Example 30 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 31 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 32 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 33 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 34 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 35 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 36 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 37 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 38 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 39 πŸ”—

Find the solution to the given IVP.

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

Answer:

\[ {y} = -4 \, t^{3} \]


Example 40 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 41 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 42 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 43 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 44 πŸ”—

Find the solution to the given IVP.

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

Answer:

\[ {y} = \sqrt{t^{2} + t + 3} \]


Example 45 πŸ”—

Find the solution to the given IVP.

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

Answer:

\[ {y} = -t^{3} \]


Example 46 πŸ”—

Find the solution to the given IVP.

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

Answer:

\[ {y} = \frac{2}{t^{2}} \]


Example 47 πŸ”—

Find the solution to the given IVP.

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

Answer:

\[ {y} = \frac{4}{t^{2}} \]


Example 48 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 49 πŸ”—

Find the solution to the given IVP.

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

Answer:

\[ {y} = \frac{2}{t^{2}} \]


Example 50 πŸ”—

Find the solution to the given IVP.

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

Answer:

\[ {y} = 4 \, t^{3} \]